Can I Have My Babys Father Pay Child Support Without Taking Him To Court?

Sure, if he agrees to do so.
The problem is, he can stop at any time, or start giving you less than what you agreed to, and you will be forced to go to court to get a child support order.
It is best for your child if you go to court and handle this by the book. You want your child to be protected by the law. If the father is trying to talk you out of it, and saying you 2 should be able to just work out a deal between yourselves, that is a red flag that he is at some point not going to pay. If you have a court order, he can have his wages garnished, his tax return seized, and, if he does not pay, he can go to jail for contempt.
If you are on any sort of public assistance, the state is going to want his info and a court ordered child support filing in order for you to get your benefits.

Yes ... But the thing is if he is a good man and a devoted father you wont have to . Some women are unrealistic, sadistic, and greedy in child support cases so talk and work with him . Also if you take him to court ,they may request less than he is offering . He may give you that and nothing more . Depending on state to state child support is from 15 to 25 % of the father`s or mother`s income .