If you are older and are re-entering the dating scene you may find you will be dating men with children. This is more and more commonplace these days, men getting awarded full custody or at least joint custody so they will have their children half of the time. This may seem like a complicated situation but it does not have to be. If you and he handle it correctly then the transition should go just fine.
The fact that he has children should be one of the first things on the list of thing to discuss when you first meet. This is not something he should hold back. If he does then you need to reconsider dating this man. His kids should come first and if he hides them from you and the world this could be a problem.
Talk to him and try to get a sense of the relationship he has with his children. Is he involved with them and their school activities, sports, and other interests? If he is truly concerned for their welfare then he is a good man and you can probably trust him with your own feelings.
Then, do a little self-analyzation. Are you capable of handling this type of relationship situation? You will be spending time with his children. Maybe not at first but at some point you will be. If they are young children then it probably will not matter if Daddy's friend comes over for dinner in the beginning but if they are a little older then they may feel some resentment toward anyone new in their father's life.
So, what's the answer here? The answer is to be patient and let things unfold naturally. Do not try to force them to like you because they won't. Children can be difficult and need to be handled with, pardon the expression, kid gloves. During this 'get to know you' period they will be relentless in trying to get rid of you. They will test you and if you fail their tests they will then go 'tattle on you' to their father.
Appreciate his situation here, too. He will be forced to be in the middle of all of this. Not that he will take sides but if things get out of control then you will see how he disciplines his children. Take note and follow suit if and when the time comes for you to be in a position where some discipline needs to be doled out by you. Children are an important part of this man's life, if you mess it up, you will be the one who will be asked to leave and not come back.
Communication is a big part of any relationship but when dating a man with children, the lines of communication have to be clear and clutter free. There can be no misunderstandings right from the start if the relationship is to have a chance at all. You may want to include the ex in the communications, too. If she sees you do not want to try to replace her then she may be willing to work with you more easily.