Hi,
by screwed I mean that if you go to the worst neighbourhoods in Lithuania, you can't expect to be safe, no matter where you're from and who you are.
In Vilnius you'll be fine, it's one of the most open and nicest cities in the Baltics. I would simply recommend staying close to old town because it would be less of an incovenience to you as you'd be able to walk everywhere. As long as you're smart and don't go out of your way looking for trouble (starting fights, shouting at people, choosing small dark streets at night instead of well lit streets full of people), you will be just fine and have a brilliant time in Vilnius.
Can't really recommend hostels or hotels, it is very dependant on your budget. Also in Vilnius I've only stayed at the 'rock & roll' hotel which was a bit pricy but very worth it. Best 3 star hotel you'll find. It is a bit further from the old town so if you don't know Vilnius, I'd recommend something closer.