There has been a lot of changes to the grappling and ground game since UFC 2, including new transitions that are only available in EA SPORTS UFC 3.
When you find yourself in mount on the bottom, the first thing you want to do is successfully blog one of your opponents strikes. Once successfully blocked, transition to half guard by pressing left or right on the right analog stick.
When using an arm trap, hold R2 and flick the right analog stick to the left or right. This is a riskier move, as if you do not execute this properly, you will get hit.
Your first line of defense when you find yourself in a clinch up against the cage is to simply deny your opponents clinch by holding the right trigger and pressing up on the right analog stick. If you fail at denying the clinch, look for a 'take down' to choose from in your grappling hud.
If you get caught in a Muay Thai clinch, you need to block a strike, and go for a transition or clinch break.
Here are more details around UFC 3 Ground and Grappling Controls. If you're starting for the first time, check out the Beginner's Guide to EA SPORTS UFC 3.