Well, we are talking about a whole century here. In one hundred years, we went from steam trains to Moon landing, from telegraph to high-speed internet. And now we have an even faster rate of change ramping up, so predictions will be even harder to get right.
There almost certainly will be a permanent Moon base and probably a Mars base, certaily more than one Mars visit. We might send a probe to the Kuiper belt, I'm not so sure about the Oort cloud. Considering there are already (very sketchy) plans for probes to Europa, I'm sure we'll have done something there. Titan might be visited, but probably not a permanent base. Interstellar probe, probably not.
Asteroid mining may have begun at least on a preliminary level. By century's end, the first child will probably have been born in space, and may be old enough to have a child or two. A few orbital space habitats will have been built, and at least one will have made a go of being materially independant of Earth (and knowing human nature, will possibly declare itself politically independant too).
On the downside, the first actual space-to-space combat will probably have taken place.
STARGAZING: All I see are the lights of a billion places I'll never go. --Howard Tayler, Schlock Mercenary