Don't worry about it
You get automatic copyright protection in the United States even if you do absolutely nothing to protect your content. If you want to be slightly more upfront about copyrights, just add explicit copyright notices on stuff you want to protect (splash screens, art, source code, large pieces of text, etc.). Something like what you see on the footer of this website: "Copyright© xda-developers. Powered by Searchlight © 2013 Axivo Inc. ". This lets everyone know that this is copyrighted and most countries will recognize it. Should you put down copyright notices? Why not, it's really easy and costs you nothing.
Copyrighting gives you some protection against IP theft should it take place (e.g.: someone stealing your art or your source code).
You can register a copyright, but it doesn't give you that much additional benefit except if there is ever a court case about copyright infridgement, in which case you can demand extra money from the other party to cover legal costs. My opinion: not worth it unless your IP starts to become valuable enough that you actually expect people to try to steal it.
In addition you can also register a trademark
. This is mainly against someone stealing your company/product/game name. Registration means that you will be listed as the rightful owner of the trademark in some government file somewhere, so if there is a dispute you can claim priority because of your previous registration. Registration takes some effort and money, and in general I wouldn't recommend it until your game/company is successful and its IP value starts to exceed the cost of registration. You don't lose anything by registering trademarks after releasing the game as opposed to before - it will be equally valid. The only problem can occur in the unlikely event somebody creates another competing product with the same name AND manages to claim a trademark it before you can file a counterclaim. This is a remote possibility and I wouldn't bother until you have a proven, successful product on your hands.