I like food... so I'll keep my suggestions focused on that.
Mexican - you can get good authentic cheap Mexican, I think around East Cambridge/Sommerville
Jamaica Plain is a cool area - and there's some good Caribbean food there (Jamaican, Puerto Rican I think). I think Sam Adams still offers tours of their brewery there.
Puerto Rican - a place called Izzy's a 10 minute walk from Central Sq. Hole in the wall, great food.
Burgers - Hahvahd Sq has a good place, all of them are named for different people. I'm not into burgers tho.
The Pourhouse - on Boylston near Mass Ave. They actually have a really good brunch.
Vietnamese - Pho Pasteur is pretty good, although the last few times I've been it's been a little uneven. Not bad, just not as stellar as I remember it being. They have 1 in Hahvahd Sq and 1 by Chinatown, maybe others.
If you like Dim Sum definitely go to Chinatown for that, I can't remember the names but there was always 2 places with huge lines.
JP Licks and Toscanini's are local ice cream chains. Not as good as Jeni's IMO, but good stuff.
North End - Italian, all kinds of places. Definitely go to Mike's Pastries for dessert.
Newberry Street is good for people watching, and there are good (but mostly expensive) restaurants.
if you are into a particular cuisine Boston is big enough to have it, although it may not be great.