Let's say you developed a time machine and you understood that you would have the ability to travel by yourself anywhere on Earth for one day to any place and time of your choosing, then when that one day was up, you'd have the ability to spend one week, somewhere else on Earth at a different time (you can't choose the same place or time). Lastly when that week was up, you'd crash land somewhere else on Earth (again a different time and place) but you knew the machine would crash and you'd have to spend the remainder of your life there never returning to the present. When I say the present, I mean close enough in time to the moment you left that you would never again be able to have anything resembling the life you have now.

1. Would you take the trip?
2. Where would you spend your day, week, and the rest of your life?

I guess that, instead of answering your question (which I myself can't very well), I'm gonna give you a gift.

It's a short story by <a href="http://en.wikipedia.org/wiki/Luis_Fernando_Verissimo">Luís Fernando Veríssimo</a>, translated freely by myself.

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Perfect Vacation

Once someone asked me what trip I'd do if there was no limit to time, transport or epoch. And budget. I concluded that no trip like this could be perfect, because, at least in my imagination, there would always be a banal detail that would ruin the rest. For instance.
I see myself in last century's Paris and, instead of a boulevard or a cabaret, I'm in a bathroom of the time, trying to get to an agreement with the plumbing. Which, as it's widely known, only started functioning well, in France, a few years ago. Of course I can't go to the boulevards and cabarets meet with the great characters of the time without taking a good bath before, even if the great character of the time don't do the same. I give up on last century's Paris.
Or I'm in the Rome of the Caesars thinking about catching a movie to rest from my walk in the Forum, and trying to resign myself to the fact that they haven't invented cinema, and even if they had done it wouldn't be of any use, because to this day there are no good cinemas in Rome and to top that everything's dubbed. Or in the flaming Florence of the Medicis struck by the certainty that the chance of there being air conditioner in the room is zero.
The ideal, of course, would be to spend a day in the Habsburgs' Vienna but only if you could sleep in the Vienna with television in the room, or make a quick escape to see Duke Ellington play in Harlem without necessarily having to stay in the New York of that time. But that wouldn't be imagination any more, it'd be delirium. I'd rather take my dream vacation in the present, with all its amenities. And I decided that my grand trip would limit itself to staying in every hotel of the world that until today I only dated from outside, thinkin about how it would be to one day come into the reception and ask for a suite, champagne in the room and alert the masseuse.

@The Gnasher

That is exactly why one would have to be careful when making their choices. Having said that experiencing turn of the century plumbing in France is extra-ordinary by present day standards because it no longer exists. It would be an event in and of itself. There is only a problem when one has an expectation of what it should be like. So the wise decision is to choose a time and place without the expectation that it would be perfect.