I think we'll need fusion to make intra-solar system travel (really) possible, where we just transport stuff through-out the entire system, with a speed of at least 1/100 of the speed of light. Achievable in ~50-100 years.
For inter-stellar space travel, say to the closest 20-100 star systems, we'll need either an anti-matter powered ship or a black-hole powered one (a bit more likely) that can travel to these stars in only a few years to a decade, and then another few years to a decade back, with the speed of 30-50 percent of the speed of light. Achievable in ~200-300 years (remember technological progress is exponential not linear).
For whole galaxy exploration we'll definitely need something like a warp engine. When we'll get this is harder to predict, but it will probably take at least 1,000 years to do it.
Things that could definitely help with space exploration: figuring out immortality against aging, which could also make us more interested in 100-year old projects, transferring our consciousness to machines/more resilient avatars not just for space travel, but also for living on new dangerous planets, and quantum Internet so we can rapidly communicate through long distances.
If we do the last one, we might not even need to travel ourselves or transfer into other bodies, to go to those places. We'd just send robots that we control absolutely and very precisely, just like we were there (Avatar/Surrogates-style).