I can't help but suggest if we had a single ship capable of 0.5c we could use it to launch lots of probes (just go top speed in a certain direction and release them). Although I guess this isn't really keeping with the spirit of the question. Next I have think we could just use it as a ferry, and set up orbiting stations around all the planets and major satellites in our solar system, which would allow parallel study of most of the solar system at once.
However, if I limit myself to only being able to visit and study anyone place at any time I'd have to say we use it for the solar system first, spending about a year here, then move on to the near stars. While visiting a completely new star would certainly be amazing I can't overlook how insanely quickly we could get around our solar system, most of which we really haven't studied up close at all (mainly the outer planets satellites). Although I suppose acceleration would greatly limit the ability to make short trips. But hey, this is fantasy anyway right?
As for if people would be willing to visit very far away places (other galaxies), I think they would. I don't think it should ever be that high a priority. But if building a large ship capable of near c speeds becomes possible for relatively cheap (and I think it will, in a few hundred years), then I don't see any reason why not. Sure the people leaving would never see the other galaxy, but they won't see it sitting here either. Their chances aren't that great, but if we never come up with some way to cheat the speed limit of light (worm holes, warp drive), which is a very likely possibility, then it means once we are capable of even 0.5c we might as well start making these long journeys. It's unlikely that we'll be gaining much more. I suppose you could argue that we should wait until we can reach a high enough speed to make the journey in a single lifetime (due to the slower passage of time for the people on board). Sure millions of years would have passed here, and millions more before any news could be transmitted back, but at least the people who set out could actually be alive for the arrival. That would also help with the problem of a generation ship losing all memory of the culture that sent it, as well as the very real problem of how to keep a ship running for millions of years.