Getting around the city by bicycle is very easy. I do it almost every day, even in winter. Capital Bikeshare, the largest bikesharing program in the country, is very popular and will be cheaper than shipping a bike if you're interested mostly in commuting. It's $75 for a year's membership, and trips under 30 minutes are free. There are stations pretty much everywhere in DC you'll want to go. I wouldn't plan on long rides on CaBi, though. They're not designed for speed.
We've had some wicked cold snaps. I bike to work in tights, wool socks, a wool base layer, a windbreaker, gloves and a balaclava. My fingers and toes get real cold by the end of my trip, but it's fine. I see lots of people wearing less.
Where are you staying? That will make a big difference in what kind of biking you can do easily. The city is great, Arlington, Alexandria, Bethesda and Silver Spring are very bike friendly and have easy access to downtown. Once you get further out it's hit or miss.