Six feet will fit most full-sized bikes, no problems there. You may find some supersport bikes too small. You didn't say what kind of bike you were interested in, but a first bike should be somewhere between 250cc and 500cc for most people, although I tend to move toward the larger of the two displacements.
If I was to make a suggestion, I'd suggest a GS500 as a good starting bike. It's not all that flashy, but they're fun to ride and have sufficient power to pull someone of your size in any situation.
Or, you might also consider something a little more sporting if that's your thing, and look at a SV650 (I'd favour the standard over the faired model) or a Ninja 650R. Stay away from new, clean bikes for your first bike... you will probably dump the bike at least once either at a standstill or at a slow speed while you're learning.
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