I find some of the answers here surprisingly dull.
If I woke up tomorrow in the 237X's, I'd explore Earth. I'd marvel at the socialist utopia it is [no poverty, no class system,..] and then I'd travel the planet. There is no way I'm going to alien worlds when Earth itself would be so alien. Of course I'd have fun seeing Scotland and how different it would be but I'd love to see what formerly 3rd world countries would be like. When I watch Trek, I like to imagine how awesome some countries must be without poverty/debt/capitalism/resource problems. Imagine a peaceful, futuristic Baghdad? A sprawling, futuristic metropolis in Addis Ababa and that's without exploring famous cities like Paris, New York, Berlin...it would be wonderful.
And no travel times! Just beam around Earth, no fear of language barriers or crime, stay in free hotels. There are so many places I would like to see NOW but to see them in the future [for free] would be wonderful. [No aeroplanes, no passports, no flights to catch and luggage to haul...]
After that [well, during that] I'd fill in the blanks that Trek leaves out. How far has technology changed? Sure I know about sonic showers but how does one brush ones teeth? I do a lot of walking, are there boots now of alien substances that are very strong/comfortable? Can I get a jacket for high altitude walking that has Andorian designs so is vastly warmer than anything today? Is there an Earth-wide computer system with every film/tv show/sport event ever made on it?
So many questions! Just exploring the technological advances would be fascinating.
Then, and only then, would I perhaps consider exploring what holiday options there are. Can I holiday on any Federation world I want? Can I just stay in a nice hotel on Vulcan/Andoria/Betazed etc and explore THEIR worlds too?
I'd be having way way too much fun exploring, experiencing [just reading all those philosophical works would keep me going for decades] and learning to even consider joining Starfleet or becoming a trader or such nonsense. The entire existence would be like being in the greatest theme park ever...and it's FREE!