You can register your name yourself online. There's a fee to register it and a fee to run a search before registering it if you want to (which is recommended to do). I've heard of people doing it themselves online but I'm lazy and paid my corporate lawyer to do it (and it was big bucks).
If you are not incorporated, then you are doing business personally that should be reported on your personal income tax return. You can have a name registered to that business. It would be a sole proprietorship which is "operating as" <NAME>.
Registering a name is a legal process, not a tax process. You still will not have to register for HST IF: 1) you weren't registered for GST and 2) you do not have gross REVENUES of $30,000. This is true whether you are a sole proprietorship or a corporation.
I'm a corporation. When I filed for a business number (so that I can file corporate taxes), I was careful not to ask for a GST number at the same time because, once you have one, I don't think you can ever go back. (Tax is not my strong point and excise tax is even more vague to me, but I don't think you can ever reverse the registration.)
I incorporated for other-than-saving-a-name reasons. If not for other reasons, I would have just kept a list of name variants and/or alternative names in case anyone ever comes forward to stop me from using that name. However, if this was my only source of income and I wanted to develop a fully fledged business out of it (even a part-time one), then darned right I would register the name!