If I only had one trip I'd do what a few of you have said and go into the far future and see what the ultimate fate of humanity is.
But, if I could go on multiple trips I'd go back and watch historical events that interest me. I'm a huge news nerd and while this might seem morbid I'd go back to see huge news events in the past. Things like go to Omaha Beach and watch the landings there on D-Day, go to Hitler's bunker and watch his final days (although I'd have to learn German obviously) Go back to New York 20 years ago and witness 9/11 live. I'm really weird in that I'm fascinated by extreme situations and disasters. Not because I enjoy seeing people die, but because I'm drawn to how people react in extreme situations like that.
Yeah, I know. I'm a weirdo.