Well, if you're interested in UK music, I can only recommend (in addition to previous recommendations):
Crass - one of my favorite bands. Possibly an acquired taste.
Culture Shock - this is the band between the punk of the Subhumans and the ska-ish music of Citizen Fish.
Exploited - death before dishonour is a great album, full of driven, aggressive, bash-the-bloke-next-to-you music.
G.B.H. - Good music with some moments of greatness.
Rudimentary Peni (I can't swear their UK, but I always think they are) - their first couple albums are great. After that, ehh.
Subhumans - one of the greatest bands ever IMO.
Also, you might look into some Oi bands, such as the Business, Four-Skins, and others. But be careful or you may end up with some racist Oi.
If you're interested in branching out to the USA, include the following:
Adolescents - OC punk, precursor to D.I.
Agent Orange - surf punk. They do a nifty version of Metallica's Seek and Destroy. Plus, IIRC, they were in an episode of 21 Jump Street.
Agnostic Front - Good hardcore music.
Bad Religion - mostly good music, although sometimes you might need a dictionary to understand some of their songs.
D.I. - OC punk, also one of my favorite bands.
Fear - obnoxious. . .but good.
Jello Biafra - this includes the Dead Kennedys, Lard, and his projects with other bands like D.O.A. and No Means No.
Minor Threat - also, some of the old State of Alert and G.I. stuff from Dischord is pretty good. Not Fugazi though! The singer's Pailhead ep is also pretty good, but more industrial.
Naked Raygun - they have some good stuff.
Screeching Weasel - they have some good stuff, and some interesting cover tunes also.
The Unseen - some guys out of Boston (I think) with some really good music.