My gut feeling is that time travel isn't going to happen.
I don't think you can travel to the future because "the future" hasn't happened yet and there's no way to say how it will play out. On the other hand, I don't think true "free will" exists, and so you probably CAN say how it would play out if you could bring all the variables into your calculations. Even at that, though, I think you'd be predicting (if accurately) something that still hasn't happened, and thus isn't "visitable".
The past is a more probable destination for time travel, but I don't think that's going to happen either. It's already happened, so at least it's static and knowable, it's a "place" if you will. As opposed to the future, which isn't. Like visiting an abandoned warehouse, versus visiting the 12th floor of an office building that hasn't been built yet.
With my *vast* knowledge of quantum whats-its, I'm perhaps not the best person to comment on parallel universes, but I don't think they exist either. Granted, you've got your particles that seem to bump into themselves, suggesting they are actually bumping into their counterparts in other universes, but that seems like a hugely complicated answer to the problem. I think the real answer is waiting to be discovered and will seem much more elegant that the parallel universe idea when we actually figure it out.
What seems most likely to me is eventually working out the impressions of the past that have been left on the universe. Every time we speak, what we say gets quieter and quieter as distance increases, until we can't hear it anymore. But does it really go away? Are those vibrations still detectable 50 years after they began? 1,000 years? More? What of the memories stored in human brains? Are they gone forever when the brain shuts down, or can they be mined from a long-dead corpse?
Those ideas, while still off the edge of believability, are more likely to go somewhere -- in my entirely uneducated opinion -- than time travel or visiting parallel universes.
Dave