What are the father's rights to a 1 yr. old baby even though the parents are not married?

The same as any other father's rights... though it most closely resembles divorced couples as far as rights and responsibilities, visitations, custody, child support etc. Marital status is not a factor in rights/responsibilities of parenthood.

Unmarried parents should get custody, visitation, and child support arrangements formalized through the courts with the help of an attorney.

Has the father or mother not already filed in family court, but they need to file for a level of custody, joint, or not, the mother if she has the baby needs to have a order of physcial custody issues she she has legal physcial custody. Then the father will want to have set visitation days or weekend and weeks in the summer set up. And who gets who on holidays.

But not being married does not take away any of his rights, it only required him to prove he is the father by a DNA test if needed.

While not having been married doesn't take away from the father's legal rights it does make it harder to enforce. You have to go to court to establish paternity, order support, order visitation, and ultimately order custody (both physical/legal). Often this takes money. Therefore it's not really as easy for a birth father to get rights established. If the mother is on any kind of state assistance it will be easier though take longer. Father can file for rights through Child Support Enforcement in the state the child resides in... The fastest is of course to hire your own attorney. Father's have rights to be an active part of their child's life, to help make decisions concerning the child's welfare, and influence how the child is raised. Though it's not fair... it's the way things are. I suppose the woman had a time of "not fair" when she had to give birth and take responsibility of a new life. Good luck. Father's should have rights without having to fight so hard. It's not nearly as hard for a woman to establish child support. Just my opinion.

If the parents are in agreement and amicable, and the father accepts paternity, it should be a simple matter of filing their agreements, getting support squared away via the state's standard formula, and not cost much money. Only when people are contentious does it become any kind of battle... and then only the lawyers really "win"

The OP didn't detail the situation, but if asking here it's probably not good.