I was in a similar situation after I got divorced. The best thing to do is to break off all unnecessary communication with your ex, and if possible, terminate communication altogether (i.e., change your phone number, e-mail address, etc.). Most divorced couples don't have "moderately friendly" relationships subsequent to the divorce, and it is usually better if there is no contact at all. It is none of your ex's business who you date now, and vice versa.
In regards to the possibility of your ex suing either of you, unless you began dating while you were married, and your ex could legitimately prove this, you have nothing worry about. Do yourself and your newly found significant other a favor and enjoy your life together. Forget about the ex, nothing but problems are going to arise from this point forward.

