It's pretty important to mention what country/state you're in when asking for business advice. We're not mind readers who know that you're in Pennsylvania or what have you
There's probably a state or federal agency where you will have to register your business... but that depends on where you are.
Here in Australia, that agency is the taxation office -- you put in your details and they give you a business number straight away, and that's it.
To trademark a name, you need to be trading a product. If you're not putting that mark on a product, then you can't claim that you own the trademark. You can put a on something if you wish to explicitly tell people that you're deliberately using it as a mark to identify your product, but it's not necessary. You can optionally register your trademark with the government (which lets you put an ® next to your mark), but this is incredibly expensive.
Copyright is automatic -- whenever you create anything, it's protected under copyright. You can put a © logo next to copyrighted material if you like to tell people that you'll sue them if they steal it, but it's not necessary.
Kentucky, USA, also would I need to copyright and trademark a game and company name? Such as if I were going to sell a game I created?