For your requirements I'd definitely recommend a decent travel zoom compact.
I've usually been impressed by the Panasonic Lumix TZ range, but haven't tried any of the newer once and from reviews it seems they've been overtaken by the competitors.
Have a look at the Canon SX260, seem to hear good things about that and it's well within your price range.
To be honest, a vacation isn't the time to start playing with the technical side of photography and more often than not I've seen first time DSLR shooters come home with terrible photos because they didn't know what they were doing.
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