If you are in the US (and this most likely applies to most countries outside the US), you already have a copyright on anything you create. But yes, you can drive the point home by putting a copyright notice on the sheet music which isn’t strictly necessary but everyone does it so you might as well. I do this as does everyone I know. It’s easy and legal though it doesn’t provide any extra protection (except maybe to discourage someone from thinking you’ve released the work to the public domain).

If you ever need to sue someone for actual money, then you need to register your sheet music with the US Copyright office. If you are planning on making money from your sheet music, this is probably a good idea.

I do not work with ASCAP or any other performance rights organizations. My music is is more in the avant-garde style of classical so I’m not really concerned with performances plus I license my music with a very liberal Creative Commons license so no one would have to pay royalties anyway. All that to say, I don’t really have any advice for that side of things.