You’re assuming that because there’s no way to travel faster than light, there’s basically no reasonable way to explore space. That’s true only for certain definitions of “reasonable”; a spacecraft capable of maintaining .01 gravities (that’s .1 m/s/s) could reach the nearest visible star system in about forty years. To get a good hundred light years out, it’s about 200 years with that acceleration. If we increase the thrusting power of the spacecraft to 1 gravity, those times start looking a lot better: three years and nine years, at least for the people on board. Granted, we have no way to get that kind of acceleration (well, continuous acceleration, anyway) right now, but it’s not theoretically impossible- we just don’t have a method to do it yet. If we make any kind of theoretical breakthrough that can allow significant continuous thrust (much better than the ion drives that we’re testing now, I mean), we’ll reach the stars. Considering the way that science keep accelerating and making new discoveries, I think it’s possible that we’ll figure out some way to do this. We certainly haven’t hit many “insurmountable” barriers that stayed insurmountable thus far.
I grant you, I’m not holding my breath either. I think we’ll get there. Eventually.
Other galaxies, though? Shoot, I’d be astounded if we ever even built a transmitter that could reasonably transmit a signal that distance without fading completely into background noise, let alone went there.