You don't want to start riding a moped if you're looking for something over 50cc -- mopeds are generally considered anything 50cc or less. You probably mean you're looking for a scooter (there is a difference.... sorry if we're assholes about it), and since you want something over 50cc, I'm guessing you want a 125-150cc scooter. In most states, the displacement and speed characteristics of a scooter like that will qualify you as a motorcycle or motor-driven cycle and obligate you to be registered and have a motorcycle license, insurance, and all of that fun stuff. Which begs the question: If you're gonna have to get a motorcycle license/insurance/registration to ride it, why limit yourself to a scooter ("moped")? For the price of a reputable scooter -- not one of the junk Chinese ones -- you can probably just get a motorcycle that would be better suited to your size. If there are any reasons why you're more attracted to the scooter/moped over a motorcycle, I'd be interested in hearing those though, since those would be important considerations in choosing a ride.
Anyway, you'll probably get some other replies here (and some of them may actually be worthwhile), but you might want to post this on a scooter forum (ScootDawg or ScooterBBS) as well. They'll have more experience with the larger displacement and modern models you're looking for (we do mostly vintage stuff here), so they'll have more pertinent advice than we will.