I'd take a Constitution class starship visually modeled after the NCC-1701-A, and incorporate the more modern technology from the Sovereign class starship design. Just for the sake of personal preference.
Using that technology I'd do a few things that people have suggested, but in a slightly different way.
Although I realize that changing history can have drastic effects on the future, I'd like to see the "what-if" scenarios that would unfold. Go back in time and ensure Hitler is dispensed with before he can rally an uprising in Germany. This would save the lives of many millions of people. But... this would mean those people get to live and have children... which would of course mean that a great many people born today might not exist at all, or in very different circumstances. For the better? I would be interesting to see what would come of it. I'm assuming it might make for a rather drastically different world. Better? I don't know. But I'd deploy special cloaked satellites as others have mentioned that would record life on Earth for the next 100 years. I'd leap forward, collect the data, make personal observations of any drastic changes, then go back in time to "undo" the change if it didn't end up much better than before. Otherwise, I'd let it be...
I would also go back in time to save President John F. Kennedy and his brother Robert. And of course, Dr. Martin Luther King. I know that would change lives too, but in a far more beneficial way, I think. I'd also find a way to prevent the uprising of mass chaos in Afghanistan that would lead to jihad fanatics, so that we wouldn't have a 9/11. I'd continue on trying to locate various events like this in the more developed nations and prevent assassinations of important people, or foil planned terrorist actions (with culprits getting caught, so at least people are aware and don't get too lax).
I'd then take a break and travel with my starship, and select crew, to visit various worlds having the potential for life and see if there really is any intelligent life out there.
Introducing starship technology to the people of Earth would be a mistake
at this time, due to the primitive levels of society (power hungry people are everywhere, ready to exploit something like this). But I'd try using the ships resources to help avert disasters on Earth, kind of like a superman... who's made of flesh but has a starship.
