[F]
Will my insurance go up in price if I make a claim for someone hitting my car and me not seeing who did it to get any information?

Do you have any witnesses?- anyone that can attest to the fact that you were indeed hit by another driver (as opposed to having damaged your own car).

If you have witnesses, then you can file under your Uninsured Motorists coverage (if you have this on your policy). If you don't, then the claim goes under Collision. How it affects your premium depends on your state's laws. Call your agent to verify the coverages on your policy and they can let you know the impact it will have on your premium.

Most likely, a few "advertise" that you will not get raised rates or cancelled for only one claim.

But if you have no idea who did it, then they will pay, and yes it counts as a claim they pay.