Fastest way to correct hifi system?

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.  

One other thought ... and perhaps better devoted to a separate thread. I am going to go out on a limb and state that I think that most audiophiles -- certainly not all -- would advise that the fastest electronic path to creating a high end system that brings you to the sound of a live event (the so called absolute sound) is to combine a tube preamplifier and a solid state amplifier.  While there may be some tube amplifiers that offer good resolution/ control at the frequency extremes, and have narrowed the gap in this respect with solid state, they are out of the reach of most of us in terms of capital. Here I'm thinking of high dollar offerings from the likes of David Berning or Victor Lamm -- in the 40-80 K range. Not to mention the cost of replacement tubes. The tube preamplifier will give your system a touch of the magic that tubes offer -- warmth, bloom, air, decay -- without a huge investment. 

+1 enginedr1960
The room is the most important, and most overlooked piece of the puzzle.
Jim Smith's book is excellent.
All of the previous posts are valid and worth paying attention to.
I've been in this hobby since 1975, and am still seeking, tweaking, changing, adjusting, and still loving it.
Don't get in a hurry, don't rush, and HAVE FUN!!!