Which has a greater influence on sound quality, the amp or the preamp?

I try not to use a component that’s obviously colored ahead of the power amps. Get a good transparent source and a good transparent preamp, then you can indulge yourself and shape the presentation a little with your choice of power amp, knowing that everything upstream is as true to the source as your budget constraints allow.

Probably the phono stage, the preamp, and the power amp; in that order. Best of the are all pure tube. 

Just wanted to totally agree with @tannoy56.

Where it starts. Source. Pre.......power and all interconnections.

Of course each step could be the weak link in a system, but you would never know how good your speakers would/could sound if you don't give it a chance to be supplied with good input.

@willyht what has your experience been?

Since I started upgrading my system from mid-fi to hi-fi 6 years ago it has been 100% Audio Research. I have been through 9 different preamps and 4 different amps. I always saw the biggest impact from amps with the preamps offering more subtle differences.  The majority of these upgrades were with the same speakers.  I tried a few different speakers along the journey but always sold the contender and stuck with the original.