We can visit up to a few tens of solar systems in a few hundred years, if we could develop a space-ship powered by a black hole, antimatter, or some advanced or innovative hybrid fusion engine (there were talks about one from the 60's around here).

Colonizing a few solar systems out of the 100 or so we can visit with such a ship that reaches a large fraction of the speed of light, should be "enough" for the next 1,000 years or so.

By year 3,000, we'll probably figure out something faster than that, to really begin exploring the whole galaxy.