Fastest way to correct hifi system?

"minorl"  Has given you some very "sound" advice.  I may throw in my 2 cents worth.  Many will emphasize loudspeakers over everything else.
It is a starting point.  I have been completely surprised at how electronics (amplifiers & preamplifiers or an integrated) can make a loudspeaker that you may not have ever appreciated, sound like you just upgraded it or even sounds completely different.  Please remember that amplifiers that essentially measure the same can sound completely different. Your ears have to make the call.
A dealer will only sell you what they carry and has a bias towards their "stuff" and may even make derogatory remarks about components that you have found to be very pleasing. 
These are important choices that you really want to get right the first time and I am not sure that there is a fast route to achieving this.
Good luck and happy hunting!

1) Invest in a good set of headphones. They will help you isolate the effects of components from speaker cables and listening space.

2) Take your time and savor the journey as much as you can. By the end, you will have a much better idea of what you like and will have developed much better listening skills. It is a joyful process.

3) Don't blow your budget on expensive cables until you have proven to yourself that YOU can hear a difference. If spending $500 on speaker wires is a better way to invest your money than getting better speakers, something is wrong.

4) Decide what media is the alpha dog in your collection. If your favorite music is all on CDs in your library, get a great CD/universal player first and hold off on a turntable & cartridge, or vice versa. Or if funds are limited at this time and you're true love is vinyl. Get a decent CD player and and hold off on the analog front end until you are thoroughly familiar with the rest of you system and have saved enough to avoid compromising your choice of turntable/arm/cartridge/phono pre and tonearm cable - the one cable that makes a big difference. IMHO

5) carefully identify 3-5 really good CDs or LPs that test a system. Well recorded with wide, deep soundstage, the vocals that really move you, the best sax solo you've ever heard, your favorite bass line, the most subtle details, etc. Use them as a uniform set of tests for all listening decisions.

All this is simpler than it sounds.

Best of luck and enjoy the journey.

I wouldn't  say this is the fastest way but the better way for better gains and satisfaction: focus on acoustics of the room.  period.   Too many audiophiles tweak their systems with expensive cables, power conditioners, and even components and speakers.  The biggest element in any system is the listening environment.  One can spend far far less on acoustical treatment that would increase the sound quality of a room exponentially more than upgrading ANY component in their system.   But it doesn't come with Bragging rights...just intelligence.

I did not take a fast pathway to purchasing a 2 channel system.  I spent 2-3 years researching, mostly reading The Absolute Sound and the advice of its editor Robert Harley, and set forth a budget (on his advisement) of 10% my annual income. Not sure it was time well spent, or a budget necessary to achieving a satisfying listening experience.   As stated above, you need to go listen, and trust your own ears.  If you want the fast approach, and are a music lover who wants very good sound to simply enjoy the music (particularly as background music), more power to you. You can achieve that with a very modest budget. I could build a very good system for 20-30% of what I spent. Take 3-5 CDs to your local audio store, or if unavailable, travel to the closest metropolitan area to visit a few stores.  Research the speakers they carry, select several you want to audition based on reviews and your budget, and listen to several over the course of an hour. Then pick a good quality integrated amplifier that will adequately drive the speakers (e.g., NAD, Yamaha, Naim, Rogue). Then pick out a source component (e.g., CD player, streaming device/DAC, turntable). 

I would suggest you audition a pair of dipole speakers (e.g., Magnepan), as they are more conducive to good room acoustics with no side wall reflections. On the flip side, they generally require quite a bit of space, and quite a few watts, to sound their best. Their sound is not for everyone, and the wife approval factor is low. 

If you want a dedicated listening room, or are an audiophile cursed with the ear of a musician like minorl, like the discerning palate of a vinophile, well then you might as well settle in for a long ride. Take your time ... again letting your own ears being the judge, and devoting an ample part of your budget to room acoustic treatments.