Welcome. I’m sure the guys will be along shortly.
For me I’d start with the speakers they will have the most noticeable change.
Pick a few you like the look of and then go and audition
I agree.
Do a little research to build a shortlist then demo them.
I'm unsure what's available where you are but here it's possible to even home loan to demo equipment which is an excellent way of finding what works in your surroundings.
Failing that most hifi stores have a welcoming demo room.
I realise the temptation to just go with whatever everyone else likes or what has reviewed well and it's a great starting place but after that were all different.
I've bought things with only reviews to go on usually prompted by big sales and mostly been lucky but it's also ended up with having to return things with all the added inconvenience thats usually attached and worse when buying second hand being stuck with it.
Foremost as "twinkletoes" said speakers make the biggest change in sound then second the amp.
The main changes are found there, after that providing the anciliries are decent it's more of a tweaking or shaping the sound even more to your liking with what you pick as the said stereo separates.
I'd go for speakers then amp and as "fandango andy" hints towards some 90s hifi is very good so maybe try a new pair of speakers with the amp you already have if it's decent
.
Obviously as CDs are primarily to your listening habits and you will need a source to demo the speakers and amp combo so using the CD player's from your shortlist to find a favourite one will speed things up.
Most hifi stores will have other things in on offer etc or recommend some unknowns that take your fancy.
After that it's just a case of getting what you want the most in order.
I'm sure many suggestions in what others like will follow to help with your shortlist, some useful some not that's where demos come in to avoid the common mistakes.
Anyway it would be nice to hear what you end up with etc and good hunting