check out modern classics (http://www.modernclassicsdc.com). their buddy models are great for the city, cute design, cheaper than a vespa, and pretty reliable (i’ve had the 125 for more than two years and 2,000 miles and haven’t had any problems ). the staff can help you find the right fit for your needs (though it sounds like a 50 would be fine for just short hops).
keep in mind you’ll need to get a motorcycle license and insurance. don’t listen to anyone who tells you that you don’t need a motorcycle license and registration for a bike whose engine is under 50cc – it’s just not true. the easiest way to get the motorcycle license is to take the motorcycle safety foundation’s basic ridercourse – offered by nova and a few other groups (cheaper at nova, but you have to travel farther and reserve your spot sooner). that will get you a waiver for the dc motorcycle road test (you’ll still have to take the written test) and a discount on your insurance.
also, get a good lock and run it through your back tire. modern classics recommended this, and i’ve found it easy to use: http://www.onguardlock.com/store/beast-8017lpt