Alright, so this is one that I see alot of people having trouble with, especially against bigger opponents, because everybody feels like they can't bridge off a bigger opponent.
Really, when you're working from the bottom of side mount, the important thing is to get your elbows and knees in so that you can get your opponent off of your hip. This is, like I said, a thread on the basics. There are some advanced escapes that you can do from here (the running man escape, the belly down escape), but really the fundamental point is to get your elbows and knees in to protect your opponents hips.
Frank Shamrock uses the term "be the ball" to describe connection your elbows and your knees, and creating that space (which protects your hips and allows you to sing back to guard) is the biggest thing.
A few little details and problems that people tend to have:
Sometimes the opponent over commits his hip and accidentally drops it on the ground. When he does that, just weave the bottom leg through and trap your opponent's upper body in guard. I land this alot because I have flexible legs.
Another thing that alot of beginning guys miss is that when you pass guard to side control you need to establish that underhook. That's really important to be aware of as the bottom guy because, if your opponent forgets that underhook, you can just shrug him forward and take the back.
Still, as far as the general side control escape (the white-belt version that also works to protect the knee on belly), it's about getting your "shields" (that connected area of your knee and elbow) between you and your opponent to keep you from getting caught in a bad position.
I'll get to joppp's in a minute.
Thanks man, very cool.