My son had a child at 15 yrs and 5 months, The mother was older and thought she
Would be hit with statutory rape so she didn't put him down on the birth certificate,
He went through delivery of the child and they stayed together for a year and a half
How can we get on the birth certificate so she can't leave state, and to where we get
vistitaion rights without a hard time, without having a lot of money?

The only way is to go to court. Now, if you don't want to use an attorney you don't have to... it's recommended but not required. The jude will have to order a DNA test and once that comes back positive he will be ordered to pay support and he can get an order for visitation.