order of improving sound quality in system?

Speakers are the most colored component in the system. Different brands/topologies sound very different, so you first need to find a pair that you like. Nothing you put upstream from your speakers will make you happy unless you have a pair of speakers that suit your tastes.

The requirements of the speakers you choose will then determine what type of amplifier you need. For instance, if you like power hungry speakers like Magnepans then you have to get a powerful amp. On the other hand if high efficiency horns are more to your liking then you need a low power SET.

There is no way of getting around this fundamental principle of building a system so the speakers must come first. You can then choose components that complement the speaker and amplifier.

I also used to adhere to the philosophy that you should start with the source, which is a very logical assumption, but after trying it both ways I came to realize that the colorations introduced by average speakers will be far greater than those introduced by average electronics.

The best speakers in the world with average electronics will still sound very, very good.

Average speakers with the best electronics in the world will sound average.

Start with the speakers.