dunno about mepis, never used it. Which desktop environment are you using ??? Gnome or KDE (or something else)?
it may be that you need to enable anti-aliasing, which supposedly smooths out the look of fonts. Or you might need to see if the mscorefonts package is available from your package managers software repository.
You might need to check to see if theres anything that will help with the fonts in the specific application that's causing you most grief i.e. firefox so check under edit>preferences>content>fonts & colours. Could it just be that you have a particularly crap font selected ???
As I say, I'm no expert but thats where I'd start checking first.