Will the cartridges sound good on the turntables ill fit them to?
Of course they will.
Which cartridge should I get based on my music preference?
It's sound preference, not music. Find a cartridge with the right sound, it will be right for everything you throw at it.
Rega leans more fast/neutral compared to say VPI which is more full/warm. Rega also are light, which is okay (where the fast sound comes from) but can be a challenge if you don't have a real solid stable rack/shelf to put it on.
Forget all the BS everyone will tell you about which cart goes with which arm goes with which turntable. First, look at output. Anything less than 0.5mV might be a good cartridge there are some good ones but that low output will make it harder for you to find a phono stage it will sound good with. Everyone focuses on cart/arm when really it should be cart/phono stage. So medium output MC.
Then read listener comments about how they sound. Because how they sound on one, is how they sound on all. People who tell you such and such works great, and that is all they say, you can safely disregard those comments. People who say such and such has terrific imaging, draws me into the music, hearing details I never dreamed were there, that is what you want to hear. Anything else, skip it. Not worth your time.
That is how you find gear (not just cartridges) that will make you happy and keep you happy a good long time.
Is my amplifier and speakers good enough to play the quality of the turntable and cartridge?
For sure your front end will be ahead of the rest of your system. But much better to have it that way, than a crap front end and everything else way better. This is another false notion, the idea that if one thing is bad it prevents hearing improvement made elsewhere. Nothing could be further from the truth. Improve your table, you will hear it. Then later on when you get a quality tube integrated, you will hear it all over again. Will you ever!
You are new so one more tip. Don't waste money on expensive setup jigs and protractors and stuff. There are free templates you can download and print that give the exact same results. Or if you don't want to mess with that then the Mobile Fidelity Geo-Disk is what I use. Then get a stylus force gauge, good quality tweezers, tiny little screwdriver, Scotch tape and a Q-tip and you are ready to mount and align like a pro.