Where do you live? In the United States, there aren't too many bikes that small except dirt bikes and some cheap Chinese imports. Still, sometimes you may run across a small Kawasaki or Suzuki in that size range.
I'm about your weight. I took the MSF class on a 125 cc Honda. It was rather cramped (I'm also 6'2"), but I could see myself running errands on this thing if I could stick to streets where traffic wasn't faster than 40 mph or so.
Right now I have a 500 cc bike. It's got enough power to go on the Interstate with no problem; I've ridden it for an hour on the freeway several times. Haven't tried carrying a passenger on it, but it feels like it should be up for that too. 500 cc's ought to be more than adequate for your kind of riding.
If you don't plan on going on the Interstate too often and do not want to carry passengers, you could get away with a 250. The Ninja 250 has enough power to more than keep up with freeway traffic too. Most of the other 250's will top out around 70-75 mph, which may or may not be enough for Interstate riding depending on how crazy your local drivers are. The only cruiser in the 250 cc class that I've found seems to fit tall riders, however, is the Hyosung / United Motors GV250. Most of the other true cruisers are uncomfortable for tall riders. The Nighthawk 250 also has room for big riders.
If you do not live in the United States, you may have that choice made for you. Some areas restrict learners to smaller bikes like 250's or even 125's. The upside of living in one of those areas is that you have a lot more options in small, economical bikes than we do here in the US.