Being a single parent is a tough job. But do not rue about it in front of your kids. They would have their own ways of analyzing the situation. Remember kids judge the situation on basis of black and white. For them there are no three ways about a situation. When they see their ideals howling, and blaming at each other they are most likely to develop fear or hatred feelings for you, and theirs is not gender biased. For them they want to see the "mommy & daddy" part together, not alternate weekend sessions.
Please do not complain in front of your kids or teenagers about your ex. Don't put such burden of disliking one of their parents on such little shoulders.
Do not introduce your kids to your new partner on some special occasion like birthday or holidays. Kids have grown up calling someone their mom and dad and the concept of prefixing "new" to it is much difficult.
Do not try to be spy on another parent, it is like the little you know more relaxed you will be. Besides, how much embarrassed you would be if your ex come to know that you are still interested in their personal details.
Don't juggle them into different sessions if they have come to you. The hobby classes and the sleepovers can wait but not kids who would want to enjoy a movie date with dad/mom. No sooner as they grow their priorities are going to change and you might be late to make up for the lost time.
Avoid seeing anyone for next few months. It will help you and your children to come up to terms with divorce.
Talk about it but don't rue about it. If it's your teenager kids, talk to them about it but do not open the intimate details of marriage in front of them, they might lost confidence in both the parents. This will also put you in confidence with them.
There are a million changes in your life. With new found freedom, a responsibility of single parent and keeping up the life standards are important. Try to work it out between the two of you on financial level so that the kids could be eliminated from the receiving end.
Keep a conversation with the ex-spouse. It will help you in future when kids would be youngsters and out open in the world and start experimenting with life. You should be in a position to talk to them and keep your views.
Children are already going through an emotional turmoil. Do not badger them more by telling them that you are taking them to new place or they may have to change school. At such delicate point, no one would like other than their close friends to witness their turmoil.