What kind of electrical projects are safe for DIY in the US?
It all depends on how much you are willing to learn. If you are confident in your ability to do it safely, go for it. If not, do more research until you are or hire an electrician.

Personally I am confident doing anything up to replacing a main electrical panel. That is something I would have to hire an electrician for, even though I could in theory do it, because you need permits and the hydro company's involvement and such.

Replacing outlets and light fixtures is pretty easy once you know how to work around electrical safely.  One small tip though - when you pull them out, but before you disconect anything - take lots of photos of how all the wires were hooked up.  Just having those photos would prevent tons of the electrical questions we get here.