Guitar amps don’t just amplify the sound, they also distort it. They just happen to distort it in a way that sounds “good” for an electric guitar. If you ever play music through one you can hear a bit better how exactly it distorts the sound.
Keyboard and bass amps give a much cleaner, undistorted sound. If you see an amp marked as a keyboard/bass amp then it has a good wide range frequency response and doesn’t distort much. If it is only listed as a bass amp, then it might just focus on the lower end and might not do so well with higher frequencies. Depending on your bass and your style of music, this might be a good thing or it might be limiting.
Getting back to the guitar amp, the first thing is to stay away from stage amps as they are going to be too overpowered for home use (you seem to realize this already). Even if you find a cheap stage amp, playing it at low volume often doesn’t sound very good.
The next thing is you really just need to hear it. Because guitar amps do distort the sound, which amp sounds “best” is purely a matter of opinion. Eddie Van Halen used to only play through Marshall amplifiers, and even then only through one specific one until it died. Tom Sholz from the group Boston rewired his amplifier until it gave him the sound he wanted. Your choice of amplifier will very much be what sounds best to you.
I personally play through a fairly clean amplifier and use a digital effects pedal that emulates many different amplifiers. Unless you get really lucky in the used section of your local music store, that’s probably not going to be an option for you. Still, it doesn’t hurt to look. My amp and the effects box were both purchased used. I have quite a few effects pedals as well. Some were purchased new, some used, and some I made myself (I’m an electrical engineer). The typical “metal” sound at the very least needs distortion. Sometimes the amp itself will give you the distortion sound (and maybe reverb), otherwise you’ll have to get an external distortion pedal. Phaser, flanger, and overdrive can also be used in different combinations. A limited budget is going to make buying these very difficult. If you can find a used digital combo effects pedal that’s probably your best bang for the buck.
I’m partial to Fender amps, but again that’s mostly a matter of opinion. Go down to ye ol local music store and see what they have. Take your own guitar with you. Don’t be afraid to try a bunch of different things out. Don’t buy anything until you like the sound that comes out of it.
I bought a lot of used stuff when I was younger (especially way back when, since I was a broke college student). While there is always a chance that used stuff will break, overall I’ve had good luck with the things I’ve purchased used.