First and foremost give yourself time to heal.
Part of that healing involves accepting responsibility for your part in the marriage not having been successful. One common mistake made by divorcees is buying into the idea that we were just married to the "wrong person". As a result, we convince ourselves that happiness lies in becoming involved with the "right" person and often prematurely move into a new relationship. Before seeking new relationships, get to know yourself again. This can be a great time to re-evaluate your life goals, as well as, what you may want in a future relationship.

Give your children time to heal.
Just as you will need time to adjust to the changes divorce brings, so will your children.  Even through you may be ready to bring someone new into your life it does not mean your children will be ready.  Be sure to give them the time and space they need to adjust to Mom and Dad being in two different homes.  

Create a supportive network for yourself.                                                   Work on developing resources and healthy adult outlets for yourself.  For example spend time with other adults you enjoy, join a book club, take up a hobby, have people in your life who will provide emotional support when life is challenging.

Don't expose your children to people you are casually dating.
It is probably in your child's best interest not to be introduced to new adults you are dating until you feel the relationship has moved beyond casual dating. It is not unusual for children to form quick attachments to new people in their lives which could expose them to another significant loss. Also, when you are casually dating someone, you can’t be certain if they are a safe person for your children to be around or if he/she will be an appropriate role model for your children.
 

Maintain parent / child boundaries
In other words, kids need to stay kids and parents need to remain the parents. Avoid using your children as your only social outlet or your new relationship committee.