* Experiment: Be willing to try some experiments. Try it the way the other parent wants it, to see if it works. Try letting the children decide how they want it to be, within reason.
* Avoid Right/wrong Discussions: Arguing about who is right or wrong will not solve anything. Instead, work on understanding what is important to each of you, then finding a way to incorporate that and resolve your differences. Focus on the problem only long enough to understand what it is, then switch the focus of your discussion to what will work, and what will solve the problem that both of you can live with your mutual decision.
* While the kids are small, still do some family activities with all of you together.
* Introduce new partners slowly and very cautiously, hopefully time will pass before you do this. Don’t spring a new partner on your spouse or your children. Don’t say “This is your new Stepmother (Stepfather).” That will set you up for disaster.
While this takes some self-control, and isn't easy, it is worth learning to handle your situation in a grown-up manner, and keep what's best for the children in mind. You'll all be a lot happier, and your kids won't be left with burdens from your mistakes.
For more information about how not to fight, see Money, Sex and Kids: Stop Fighting about the Three Things That Can Ruin Your Marriage.