1) The "bad" part of town is called South East (SE). You generally won't ever have a reason to go here anyway unless you are going to a Nationals game.
2) Everywhere else, especially NW, is usually pretty safe. National Mall doubly so.
3) Don't go jogging in Rock Creek Park at night. Or just be there in general at night.
4) Metro can be your best friend. If you are concerned about a thief stealing, say, your phone, then just leave it in your pocket. Additionally, Uber is about as solid as can be in DC and the surrounding areas. If you are ever considering taking a taxi, don't even bother (they're horrible). Uber 100%. It's cheaper, cleaner, safer, and the drivers are usually fun to talk to (as opposed to the sad life stories of cabbies). Alternatively, I hear Lyft is pretty big around here as well.
5) Not to be controversial, but if you are a lady, I would expect you probably already know some of the basic ground rules for staying safe. If you're going somewhere you don't know well, travel with a friend or a group, maintain awareness of what's around you, stay in places with other people, etc.
6) Last thing, if you ever get someone accosting you, either by eyeballing you, saying something, just pretend as if they don't exist and keep walking. One problem DC has I will mention is, especially nearer to the Capitol, you'll get a lot of homeless standing around outside the metro.
7) On another note: whenever you go into a metro in the morning, you can expect one or two guys to try to force newspapers into your hands. One is produced by the Washington Post which is okay, the other called the Examiner is owned by a conservative billionaire who tries to push Obama-is-a-gay-Kenyan-Muslim conspiracy theories. Better just to avoid these altogether (unless you like crossword puzzles).
DC is a great city and just FYI, it's come a long way since crack was going on back in the 90s. If you use your common sense, you'll be perfectly fine. Oh and like the other guy said, don't stand on the left side of the metro or I'll shoot eyelasers into the back of your neck.