I leave to the army soon so I try to spend as much time with him as possible but its very difficult because our son lives with her. Im scared I might not have my son to come home to.
Our relationship gets rocky at points and since our son lives with her when I get to visit him its only for a hour or 2.
She has filed child support on me because her mom talked her into it and I buy him whatever it is that they need.
I want to have some type of custody rights to him. What should I do?

You to court and apply for custody. Unless there is a good reason, courts will usually award joint custody between the parents, although one parent (in this case the mother) will have more time spent with the child, both parents are responsible for the child, and have the right to quality time with him.

Start paying support. Your son needs consistent support, and buying things now and again doesn't cut it. We're talking money to pay the electricity bill, food, gas, school or daycare fees, all that stuff. Pay regularly, and pay enough.

She lives rent free with her mother and I asked her to come move in with me for the last remainding months that I'm home but she said no.
Day care is coming up and I'm going to pay my half.
And there talking about raising the childsupport when I left to bootcamp.

It makes sense for her to live with her mother, raising a small child without a husband is difficult, she's going to need all the back-up she can get. You're not going to be there.

Don't just pay half of childcare, set a regular amount, and pay that every week or every month. Not only is it good for your child, it puts you in a much better position when you go to court to look for joint custody.

First you have a child, and that is not a reason that the mother has to be with you. You need to work on being able to see the child and this may normally mean not at the child's mothers home.

You file for either joint custody or at least visitation though the courts , just like she filed for support.

And yes, support will be passed on income, so if your army pay is more than your old pay, it would increase.

This is a legal issue and has been moved from Parenting.

As noted you NEED to go to court and establish your custody rights. If they have filed for child support you counter file for custody. You have to establish your rights in court.