Looking to build a system for under $5k

There is a lot of good advice here.  I didn't read anything that would be really bad advice.  However, there are a couple of key considerations in your case.  This is your first serious system, and you are working with a fairly low budget.  Therefore, I suggest that you look to the future.  In other words, this won't be your final, maybe not even a long term, system.  I would try to get one or two components out of this that will serve you well for years to come. The other thing I have noted is that you are going to be using a small bedroom.  In view of that later consideration, I would avoid speakers that go much below about 40Hz.  Speakers that go low can be very difficult to integrate into a small room.  You can have really good speakers sound really bad in a room like yours because of too much ringing in the low frequencies. You may also want to eliminate speakers that need a lot of space between them and room boundaries.   Given this, I would probably forget about panel (dipole) speakers.   There are some good relatively inexpensive  monitors that might work very nicely in your room.  It is important to hear speakers before buying.  If there are some good salons within driving distance, spend some time finding a speaker that you like that will work well in your room.  Try to negotiate an in home audition.   Your speaker decision will dictate where you need to be looking with respect to an integrated amp.  Some speakers like tubes, some almost demand solid state.   The Coincident Dynamo mentioned by Charles is a superb value and can take you years down the road.  But, you really are going to need a high efficiency (~92 to 94 dB) high impedance (8 Ohm) speaker.  I use a dynamo in my second system. It is a really nice little amp.  Both the Marantz 8005 and Sony HAPZ1 have been mentioned earlier.  I haven't heard the Marantz, but I can tell you the Sony is superb.  I had my Sony modded by Dan Wright, but it was a superb value at 2K before modification.  Both the Sony and Marantz offer a serious upgrade path by sending them to ModWright down the road.   There is no doubt, you can get a very nice little system for 5K.  Just be mindful of the limitations imposed by your room, and you should be fine.

What I tell beginners looking for speakers is, to look for efficient ones. In other words- speakers that are easy to drive, as this greatly increases the choice of amplifiers that can be used.
Good luck and have fun buying your new system,Tish

Racamuti,
Good and wise advice. There terrific low and moderate power amplifiers available  that truly  thrive with this type of speaker. 
Charles 

Here's a good speaker match for your small size room. It is also a component you can hang on to for years to come and upgrade around.
It is a little more then you probably want to spend but I think there is some price negotiation to be had here, since I feel the asking price is a little high. They have a sensitivity of 92dB, and a Impedance of 8 ohms, . That's much higher then almost all stand mounts and will match with the single ended and low power push pull amps mentioned above, although I would consider a Primaluna Prologue Premium integrated.  I have listed a link to one for sale, where I think there is some room for negotiation.  Down the road you can upgrade the sound quality of the PL with upgrades in tubes / fuses and a PC. It's obvious that you would have to spend probably another 1K above your budget, but since these components are available used, your getting a lot of value and sound quality for your money.  Both the speaker and integrated are very well built.  The Primaluna is all point to point wiring, ececpt for a board or two for auto bias and protection circuits.
There are many reviews on both the components listed below; I would highly encourage you to read them.

https://www.audiogon.com/listings/monitors-reference-3a-mm-de-capo-i-beryllium-2017-01-22-speakers-3...
Note: you can get spikes for the Sound Anchor Stands through Sound Anchor (husband and wife team)
Note: They employ beryllium tweeters.

http://www.usaudiomart.com/details/649291245-prologue-premium-integrated-amplifier-dealer-demo-unit/

As far as sources, the Marantz and Oppo players outlined in above post would be very good choices for around 1k

Reference 3A MM Decapo BE (with sound Anchor Stands) = $2,400.00

Primaluna Prologue Premium Integrated = $1,850.00

Clearday silver biwire cables (new) = $350.00

Reasonably good interconnect on used market used =  $300.00

New Marantz SA 8004/8005  or New Oppo 105  Multi CD player = approx. $1,100.00

Total $6,000.00

Not trying to debate @douglas_schroeder audio show recommendation, as that is indeed a great opportunity to get  firsthand experience with a lot of different systems and approaches. However, with a few notable exceptions, I've heard/seen very little that can be had for a lower budget: Van Alstine components (paired with $9K Salk speakers), Odyssey Audio, and Fritz speakers come to mind. That was pretty much it. At the last show I attended, there weren't any other speakers on hand for around $2K or less besides Fritz ... none that I can think of right now. So besides the aforementioned companies, what shows are out there that feature lower priced quality audio products?
 
It appears that the average audio show system price is easily north of $20-$25K, with some in the $100's of thousands. My system was put together from online research, studying, and auditioning gear in my own home. There was a fair amount of trial and error, shipping back and forth, buying and re-selling (used gear ... sometimes for a loss); but eventually I got there. My recommendations represent a kind of "if I knew then what I know now" approach.

So, what's the point? I almost dreaded going to that first audio show. Actually I was worried that - given the somewhat addictive nature of this hobby - I might leave and feel like taking out a 2nd mortgage once I got home. My actual experience was the opposite of what I expected. Maybe it's just me, but a lot of what I heard at those shows (and from some very high-end, high dollar systems) sounded like exceptional "hi-fi". Very little sounded like music. Following a few such experiences, I became more satisfied with my own rig. Go figure.

Note: I too, will not comment further the merits of audio shows. This is just an account of my personal experience. If someone else wants to weigh in, great ... but that's it for me.