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. You can also register copyrights with the government, but it's also not necessary unless you're suing someone.