IME it's not a good idea to draw general conclusions from a single first-hand experience, especially where audio is concerned - it's something we all do (I'm certainly guilty of it!) and is a fairly fundamental human trait, in all areas of life, but I think it can be detrimental in this case. Speakers and amps vary
so much that there just can't be any generalisations made. In some cases, budget amp A will drive expensive speakers B just fine (in terms of power), in other cases, with other combinations of amps and speakers, the same may not be true.
In my experience, speakers tend to make the biggest improvement of any component in the hifi chain. Generally, in a system with £300 components, jumping to £1000 speakers will make much more difference than jumping to a £1000 CD player or amp. But again, this is a generalisation, and I would never claim that this is always going to be true.
It's definitely true that most speakers will almost always benefit from an upgraded amp. The degree of that benefit depends on a lot of factors, of course.
Personally, I think upgrading speakers first is a fine plan. It will
most likely lead to a big improvement. Then, from there you can upgrade the amp (and likely see a big improvement as the high quality speakers are used to their fullest - probably much bigger than if you'd upgraded the amp before getting the nicer speakers). Then the source can be upgraded and you'll have a much better chance of noticing differnces in source quality than if you'd upgraded the source with the old speaker and amp.
Anyway, just my opinion, and I readily admit that everyone's circumstance will be different and everyone's system will be different. As always (it's getting boring having to say this yet again

) you have to try things out for yourself and see what works best in your specific case. No sense going on broad generalisations.
Cheers,
Dunc