The keyword would be JURISDICTION.
If you have firmly decided to pursue custody, do not let the jurisdiction go out-of-state.
The disadvantages of distance are immense: travel, lodging, and the difficulty of seeing and communicating with your child.
Your girlfriend may eventually file for child support later (6 months+) in the new state, so the jurisdiction ONLY AT THIS TIME is up to you.
Once she is served in your home state, generally she will encounter compelling reasons (legal ones) to stay for a while…
Use this time to devise strategy, to negotiate in good faith with the mother, and most importantly, to establish yourself as THE DAD in your child’s life.
Good luck, whatever you decide to do, and welcome to the club…