You should register the brand name. It may not even be necessary to register the logo or symbol, but you could always do it later on if you really want to.
Typically, copyrights are for works of art and trademarks are for logos, symbols and brand names. That said, if you want to consider your logo or symbol a work of art, then you technically can copyright it. But it wouldn't grant you the legal protection and recourse you are probably expecting.
Long story short, copyright is not a good alternative to trademark.