Submissions on Pride FC are pulled off like throws in Tekken. You either press LP + LK, or RP + RK simultaneously to execute the Sub, GNP or switch/sweep depending on the character and position. Now, in order to escape from any of the grapple attackes the general rule of thumb is this:
Right as a sub/grapple is being started you can escape by inputting the appropriate command. For any Chokes or Armlocks, hit both punches (LP + RP) during the initial escape window (more on that below). For any leg locks and leg takedowns, hit both kicks (LK + RK) during the escape window. If you do it early enough, you won't just escape the sub, but you'll also improve your position (sometimes passing guard and going to full mount, sweeping, etc). Do it in later in the window, and you'll just do enough to fight off the sub, but you won't better your position.
Now the escape window is based off of your Characters "Submissions" stat and their current amount of Endurance (depicted by the blue bar in your health gauge). If your submissions stat is high, you'll have a longer buffer window than if it was set low. Also, if your Endurance is high, you'll have a longer window to "fight" out of a submission (see below). This makes it so that if you wear down your opponent, it will be easier to catch them in a sub as the buffer window will be much smaller.
If you hit the appropriate button to escape, but it is very late in the buffer window, sometimes you'll find yourself stuck in a stalemate and you need to fight out of the submission hold. This is accomplished simply by thrashing on the buttons as fast as possible and controller spinning (doing spinning motions with the Left analog stick as quickly as possible). If you do it fast enough, you'll fight off the sub just enough to escape it but sometimes you'll be left in a worst position than before. Being able to fight off a Sub is more dependent on your current amount of Endurance, than it is your Submissions rating.
Some characters have specific escapes and counters using the Left Grapple (LP + LK), Right Grapple (RP + RK) and cross Grapple (LP + RK or RP + LK) commands against specific types of attacks. For instance, Skauraba can take an opponent's back if they shoot (LP + LK), catch them in a suplex (RP + RK) or roll into a Kimura (LP + RK) depending on the counter command used. You can find these by going through the training mode but some are listed in Pride FC FAQ page on here on Game Faqs
If you need any other info, feel free to ask. It's been forever since I've played the game, and I know I wrote up something on it for another forum I used to frequent. I don't know if I still have the post saved anywhere though.