System Synergy

My experience has lead me to believe the best way to go about it is to find speakers that you like, but that also work in your room. Speaker to room matching is the most important thing to get right. Room treatments are critical here.
Then find an amp (integrated in my case, preamp + amp for others) that works well with those speakers.
Then the sources, then cables.

I say this because I've had some really good speakers with some really good amplifiers, but because the speakers didn't work in my room it was all for naught.

Technically it's matching impedances of source -> amp/pre -> speakers.
The general rule is where output impedance is substantially lower than input impedance.

I agree with Byroncunningham's suggestions. Start with types of music you want to play first and speakers that can play those types well, then select what will work in your room from that group.

I also agree with Byroncunningham's observations except that I think it omits an important initial step, that is determining which type of speaker (dynamic, horn, planer, electrostat) is most likely to produce the type of sound and 'sound-stage' that is most important to you. They all do it differently and what I prize may not float your boat at all. You must also assess your willingness to really work to get the best results out of the type you do chose.

Most high quality results do not just happen when you follow generic advise from reviewers and audiophiles. Formulas, to the extent they exist at all, are merely starting points and are devoid of sensitivity to personal desires/needs.

FWIW, it took me many years just to find the type of speaker that I could make work for me (that was before the Internet existed) and once determined it took many more to find the specific speaker that did exactly what I expected out of a 2 channel system set up to excel with classical and jazz music.

Marakanetz has the "long term solution". Do I not buy the $3000 Paradigm towers I've been salivating over because they might not sound good in the room? No--to use a double negative. The room is more or less acousticly friendly to sound which comes from all speakers. I know that wood, carpet, glass, concrete, bookcases, drapes all contribute to the acoustics, and I probably know how to improve the acoustics. The only question I ask in regard to the room is whether or not it is too large or too small for the speakers. Component matching is the great challenge in system building and the fun. Maybe amp A is better than amp B. But if I can match amp B with speakers that it is not underdamping or overdamping but damping just right, I can get it to sound better than amp A which is underdamping the speakers. The room cannot correct an amp-speaker mismatch and that is also why Paul Speltz's autoformers would be a fun toy to play with. If you buy companion separates or an integrated, part of the matching has probably already been done by the designer, but you still have the source and the speakers to match. I'm an audiophile stuck with a liberal arts background so I'm not very articulate on the technical aspects although many members could articulate the details of matching components. But the technical answers are the long term solution. I think personal preferences follow. I don't think one says, "wow. I really prefer the sound of an underdamped speaker to one that is perfectly damped".