There are two kinds of punk rhythms: slow and fast. You'll definitely notice the difference if you hear them; one sounds like noise produced by psychotics, and the other sounds like noise produced by hyperactive psychotics. I'll focus on the fast ones here.
It's pretty simple. The basic rhythm is bass on the downbeat, snare on the and, and high hat on all eighths. Typically, only the bass line is embellished. Consider a pretty dang fast tempo to be 220bpm. Yuri Ruley is said to play beats like this at 250bpm. I don't have my verification hat on today (nor any MxPx albums nearby) so I will rely on Internet hearsay.
Here's a version with typical basswork thrown in. Each column is a 16th. Play it at around 150bpm and you've got the basic feel.
Hh|x| |x| |x| |x| |x| |x| |x| |x| |
Sd| | |x| | | |x| | | |x| | | |x| |
Bd|x| | | |x|x| |x| |x| | |x|x| | |
|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|
1 + * + 2 + * + 3 + * + 4 + * +
The bass line is usually played with only one pedal. I recommend adjusting your pedal for bounciness. If you want to take it up to extremely high tempos, a double bass pedal will help keep the bass part tight and accented. At these speeds it becomes harder to consistently elaborate on the bass part, so it's often reduced to the simpler:
Hh|x| |x| |x| |x| |x| |x| |x| |x| |
Sd| | |x| | | |x| | | |x| | | |x| |
Bd|x| | | |x|x| | |x| | | |x|x| | |
|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|
1 + * + 2 + * + 3 + * + 4 + * +
If you don't play the bass part on this line extremely tightly, then (when you become a famous punk drummer) prepare to be derided on web forums the world over.
Thanks be to: my lack of drum skillz, Lagwagon, BeerBong, Black Flag, Anti-Flag, MxPx, Bouncing Souls, and many other crazy-ass punk bands.