DC will be out of the question. If you want to live in a safe area, you would struggle to find a 1 bedroom in that price range.
Silver Spring, MD is worth looking into. It's not far from Bethesda. The Downtown portion has many nice apartment buildings and is very safe yet urban and walkable. It also has a DC Metro stop, so you could hop on to do things on the weekend in DC. Finding a 2 bedroom in your price range might take some looking, but is doable if you find the right kind of move-in special.
You might consider looking north into MD, like Columbia, MD. It's a little drive north, but it is far more affordable that the immediate DC area and is very safe. You'll find the area is comprised almost entirely of families. The commute down 29 is not terrible. The schools in Howard county are some of the best in the nation, which will be a concern before you know it.
Here's a list of resources that might help:
1)
Craigslist is a great resource for apartments in and around DC, but it's overwhelming if you don't know where you want to live. Start the process by determining the AREAS you like, then look for apartments in those areas. The search function and price limitations on the site are your friends, but only once you have your bearings.
2)
SpotCrime allows you to plot different types of crime on a map so you can see what areas are safe and which are shady. You'll see some very clear trends in DC. I focus mainly on shootings, burglarly, and robbery. Assault is most commonly against people the assailant knows. Theft can happen anywhere.
3) Once you've got some areas in mind,
HotPads plots out apartments from various sources on a map. The Google Real Estate Search tool on Google Maps (click My Maps) is decent, too.
4) Google Maps is a FANTASTIC resource for looking into areas. If you're interested in living in DC and want to use the Metro, click on "My Maps" to get the DCist Metro Map overlay. Then, use walking directions or the distance measurement tool to see how far an address is from a metro. 1/3 mile (1760 feet) is the ideal range I'd look for if possible. It's a short walk, and is quite manageable even in bad weather.
5) Use Google Street View to check out the neighborhood around a building to make sure it's a well-kept area. I'd pass on a nice building across the street from a boarded up one. Also, look at the parked cars. Clean, newer model cars tend to speak well for an area. You don't have to look for Lexus logos, but a trend of run-down beat-up cars may belong to residents that don't take care of their stuff and probably won't respect YOUR stuff. (Nothing to do with race or class-warfare. Just a fact of life. I'm lower-middle class, of a darker persuasion, and drive a Ford.)
6) ApartmentRatings.com helps you find buildings that LOOK nice and may have nice "model/demo apartments" but have serious management, pest, or other problems. See what former/current residents have said. It's not a site to find your apartment on, but rather it's a site to check on a building that you might be interested in.
7) Look into what's near an address using Google Maps. It's nice to have a grocery store, dry cleaners, pharmacy, etc nearby. It's even better to have it within an easy walking distance. DC is one of the most walkable cities in America - take advantage of it!