Your Mac Mini is capable of being upgraded to Yosemite (free in the Mac App Store). The specified minimum of RAM is 2GB but you are likely to find it runs very slowly on that, with a lot of beachballing. I don't know what the maximum RAM is but you should consider installing extra if you can. Even so you may well find it's rather slow - Yosemite is really optimized for modern Macs running SSDs.
Snow Leopard isn't getting any more security updates by the look of things, so there's a lot to be said for upgrading it. However if you install the latest FireFox (36.0.4) that will run on Snow Leopard and should at least solve the inaccessible sites problem.
If you do decide to go for Yosemite you should make comprehensive backups: I would advise if possible cloning the entire Mac to an external hard disk, so that if you find you don't like Yosemite you can clone back and be no worse off. Otherwise downgrading is frankly a pain to do.