Building a New Listening Room on a Budget

My wife and I have been considering our next home, and recently signed a PA agreement with our builder.
The basic layout of the house is set but not finalized, which should be done within the next week.
I will have a dedicated room for my audio needs.  It measures approximately 25' x 14', and is not a perfect rectangle.  Doing my best to describe it, there is a 14' x 14' square section on one end and a 11' x 9' section on the other.
I plan to set up in the larger section, and use the smaller section for record/music storage.
The utilities room for the house is in the adjacent room, and I plan to have the internet router in that room and run a LAN cable from there into my listening room for streaming purposes.  I already plan to have at least one dedicated power line installed.  Should I add a second?  And, should I do 20 amp or 15 amp?
I'm planning to insulate the ceiling and walls adjacent to other rooms inside the house to avoid disturbing my family with the noise.  The floors will have carpet and walls will be traditional drywall.  
Beyond these things, what should I focus on that will not cost a ton?  I have read Robert Harley's article on his dream audio room build, but he spent way more that I'm willing to.  If I had to put a limit on it I would say around $10K.  
What other areas should I focus on?  
Thanks in advance!
Each room may sound slightly different (before or after room treatment)

You want the path from the speakers to your ears to be as unhampered as possible. In other words, are sound waves going to bounce off of other objects in the room? If so, the first step would be acoustic foam or acoustic vinyl barriers to mitigate any undesirable effects of the room. Ideally, you want to create a private space for the system.

Carpet is good for audio gear! And with regards to 15 A or 20 A, do you see yourself using power amplifiers/mono blocks in the future? If that's a yes, then go for 20 A. Always better to have more power on tap - just in case! 
Think about what things are easy to do now but would be extremely difficult later after the room is constructed.  Not all of these things are way expensive.  You can always buy new gear later but if you are going to stay awhile there will probably be some useful suggestions by the folks here on what you can do to the room now that will provide flexibility for the future.
  • Insulation in the walls, ceiling (what is above?), type/thickness of drywall used - not making recommendations just pointing out this stuff would be difficult/impractical to handle later
  • Cable routing - consider where your electronics will be located and where your speakers will be located and then decide whether it is ok to have wires on the floor or whether you want to build some sort of internal routing.  For example if you plan to have HT speakers in the ceiling, or in the back corners of the room, now is the time to run those wires.  If you are considering a swarm subwoofer system, do you have provisions for IC and power to those rear subs without wires across the floor?  Also, if your speaker wires must pass by a doorway or maybe a fireplace, now is the time to consider some sort of conduit(s) that can carry those wires under the threshold or through the fireplace and allow for future wire changes.
  • Power - 
  • When considering outlets, think about where your equipment will be located, i.e., will the electronics be on one wall and speakers on another?  Might you want monoblocks near your speakers, and therefore need power outlets near where speakers will be located and away from the other electronics?  Will you possibly have HT equipment that you want to locate and power separately.  
  • When considering power, there are some here that buy into the viewpoint of having only one power line for everything and that solution works for them.  OTOH, I have had absolutely no noise issues whatsoever with multiple lines that are all run off the same side of the main electrical panel.  I run high powered amplifiers that each have their own 20A line.  I have a third 20A line for the electronics.  Nothing else is powered by those lines - the wires run directly from the main panel.   In addition to the audio circuits, have your electrician install his regular circuits for lighting and other normal household electronics.
  • 20A circuits are probably most common for those who run dedicated lines.  Like many here, I upsized the in-wall wire for my 20A circuits to 10 awg size, which your electrician will tell you is only necessary for 30A circuits.  Some here use special "audiophile" wire and there are places that cryo treat romex, if that stuff is important to you.
  • Consider your situation, and the location of your audio room relative to the main panel (although you indicated it would be close), and whether you might want to set up a sub-panel nearer to your listening room and have all the audio lines run directly from that - that would be a discussion with your electrician
  • Some here have promoted the benefits of whole house surge protection that your electrician can install now and some use large filtration units for the audio related lines.  
  • Do your stream music or have a server?  If a server, do you want that in the utility room? Do you need to route wires and power the server?
  • Is this the lower level with a floor slab-on-grade, or a suspended floor on wooden joists.  If on joists, do you want additional blocking, reinforcing, subfloor thickness, or even deeper joists to reduce deflections, or maybe just reinforcement beneath where your (large?) speakers will be located, or below your turntable (if you have one).
  • Some people have had good luck running fiber cables from their router to their digital audio equipment.  Now would be a good time to run fiber in addition to your LAN cable, if that is something you might consider using in the future.  I have both and don't really hear a difference yet but need to listen more.  Fiber cables are cheap so running one in addition to a CAT LAN cable would not cost a lot.  Consider which CAT series cable you want - there are threads here and other articles discussing pros and cons, which mostly have to do with different levels of shielding.
These are the main issues I can think of based on my new construction experience and the things I have had to do or wanted to do later.  I am sure others who have built rooms can add to it.
I was really... really lucky when we bought our house... I accidentally chose a house with a near perfect listening area. My high audio guy says he has never heard one as good... and he has decades of experience installing many $200K systems. So, let me tell you what I think are some of the features which helped it.
First it is a daylight basement. The room is not a room but an area within a large area of an open plan and the system is set up against a wall that behind it is underground and is a solid concrete... dead silent. Also twenty five feet or so, left or right it is also underground... Dead silent. The downstairs is a sprawling Z shape with the audio system in the center of the Z.  Isolation is created by it being down stairs and the living area is upstairs. The center audio area is 25’ by 35". Behind me is my library... open passage 15’ wide...then goes off another 15’ feet away from the audio room. To my left in the center is a hallway away to a couple other rooms. The space is rife with bump outs (bar... closet under stair)... only the wall with the speakers is straight. The ceiling is 8’ or whatever is standard. Above my seated head is a lowering of the ceiling by a foot... perhaps for the air conditioning... it is finished. I am not saying this to help you duplicate it. But that the key is quiet and asymmetry in all places except the fromt wall.
Also, my partner is disabled and has real sleeping problems. The bedroom is upstairs and on the other end of the house, so I can turn my system to any volume and it does not ever wake her.
Carpeted floor. I think the behind me being open and a long way to the back outside wall is an asset as no reflection.

Put dedicated lines on all three walls near your expected system location. You are sure to decide to move it at some point. It is really cheap to do now. If you are sure... put two outlets. Make sure the electrician wires them on adjacent breakers... putting them in adjacent breaker boxes can cause a ground loop between the outlets.
If this site has PM. I’ll be happy to send you some photos, them may make it easier to understand.
If I was building a room, and didn’t know better... I would have built a "room".. but a sprawling area is much better!
10K, or WAY less, I’d say. 1 20 amp dedicated and an outlet every 4 feet @ least two different circuits on the SAME rail at the mains.. Three total 1 20 dedicated..

I’m not much of a studio look kind of guy.. I use HEAVY retractable curtains and I’m working on Spring bass traps.. Mainly to cut down on the studio look of clutter. I would stay away from foams most of them are OK at best acoustically. Square traps are the better of that type of bass trap..

Sure nice when a plan comes together. 35 year on mine, I’m just now able to finish the work after adding 4 rooms AFTER work.. Very little help.. Just me my hammer my saw...NO NAIL GUNS either. Double dip galvanized nails too.

Sheet rock was 5/8 fire rock on the lids and 1/2 on the 2 x 6" wall studs.. I had help on the lids a kid help me.. Everything else I bucked up.. I was a BIG guy back then. I worked in the nude too.. Just a nail pouch and work boots..:-)

I glued and nailed the T&G 3/4 ply too, NO NOISE in that room..

The second room has a built in 128CF bass traps on one end of the room.. I’m droolin’ to finish that one.. open the traps and WALA perfect room almost.. lol It’s also my indoor work shop too..
GEAR everywhere. :-)

On that kind of budget I would do only the few things that make a huge improvement for almost no additional cost: 

IF you sheetrock anything, 5/8" blocks sound about 15dB better than 1/2" but costs almost nothing more.  
Use a solid core exterior type door. You can buy architecturally designed interior doors made for sound abatement, but they can cost a small fortune and be hardly any better than an ordinary solid core or exterior door. The exterior part matters because you want weather stripping. This one thing right here costs very little but makes a huge difference. 

ONE SINGLE 20A line.

That's it. The only construction I would do is if your layout makes it cheap and effective to add an interior wall for sound control. The space between could then become a bass trap. But you have to find someone who understands acoustics and balance that with your floor plan and use.  

Honestly from everything I have seen going from planning to construction and then ongoing system improvement, I know everyone loves to go on and on about the room but my experience is you get a lot more bang for your buck from things like Synergistic HFT, Townshend Podiums, Cable Elevators, and wire, than anything you can do to a room for the same amount of money. 

So not saying don't spend money on the room. Those few things I mentioned are huge. Mega. But anything beyond that, if it means not having money to do these other tweaks, then you have shot yourself in the foot, committed own goal, and put the cart before the horse.
Thanks to all for your responses.  Very helpful!
Ruminating on a few things:
- Stupid question:  Can 15A components be plugged into 20A outlets without a power manager?  I have a Niagara 7000 and know that it is fine with 20A or 15A, just not sure about plugging in directly to these outlets.
- The room is in the basement, partially underground.  Two walls are outside walls, brick covered with studs and drywall.  One of these walls will have an egress window, the other will have only dirt behind it.  The other two walls will be barriers between inside rooms of the house, one of them being our family room, and the other being the utilities room.  I'm thinking to insulate both of these walls with rolled insulation.  I could spray foam them at higher cost but wondering if it would have any benefits?  Ceiling will also have rolled insulation unless I hear of a better way.  BTW, above this room is the walk-in closet of our master BR.  I do not want my wife to complain about the noise at night.  At least it's not directly below the sleeping area but we all know how annoying it can be to deal with noisy neighbors in apartment settings.  That low rumble that persists.  Don't want that.
- 5/8" rock is a great suggestion.  I already requested this with the builder.  Sheetrock guys will probably curse me.  
- In my current room I have the equipment rack to the side of the speakers rather than between them.  I have heard this produces better sound but not sure I can attest to this.  I do have the option of running it this way in the new room as my speaker cables are 16 feet long.  Not sure if this is important to consider but thought I would elaborate.

Thank you!

A lot of good tips mentioned here I wished I knew before I got started on my room on a budget. I particularly like the idea of using thicker insulation, carpeted flooring, the 5/8" sheetrock is a great suggestion, run plenty of cables, use a more insulated heavy duty door with foam seal all around to keep more sound in. For the walls, if you can meet the ceiling at a 45 degree angle it would eliminate a lot of in room corners that is usually responsible for excessive bass.
Wall treatment is a must in a dedicated room! more parallel walls to deal with than in conventional living rooms. Stay away from acoustic foam or use very sparingly, a dead sounding room is the last thing you want.   
Below are some links you might find interesting wrt your proposed room.  
The electricity delivery articles offer similar recommendations with a few specific recommendations that vary somewhat between articles. 
To the question of one vs. two or more dedicated lines, most of the linked recommendations address multiple dedicated lines.  Michael Fremer from Stereophile recommends only one line to power the whole system to reduce the risk of ground loops but the linked articles do not necessarily support that recommendation.  IME, you shouldn't have ground loop issues if all the dedicated lines are of similar length and all terminate on the same side of the electrical panel.  My 3 dedicated, 20A lines are dead quiet.  You need to make that choice carefully as it would be hard to add new lines later.  You have one chance to easily construct and finish your room during the initial construction so I suggest looking at the articles, and considering the following when deciding how to wire the room;
  • your system's physical layout and potential future changes (i.e., front end components on a side wall away from the amplifiers, monoblocks near the speakers, etc.), 
  • power needs of your gear and particularly large power amplifiers (look at the  DTCD — Dynamic Transient Current Delivery thoughts in the Galen Carol article), and 
  • whether you might want certain types of gear on a different circuit (i.e., in the event you want to use a conditioner on some gear but not all)
Definitely have the electrician run a different circuit (separate from the audio gear) for general use outlets for things like lights, televisions, where you plug in your computer, etc. Also, consider a whole house surge protector.  If the audio lines will be significantly different lengths, you may think about setting up a subpanel and then running only the audio lines from that but you can have that discussion with the electrician.  Links below: