Chandler, I had many of the same concerns. I am moving to Philadelphia and just spent the last 3 days visiting apartments and exploring the areas.
I think you will be safe as long as you take precautions. I think most places, if not all, are pretty safe during the day. During the night, it is best to avoid certain areas (the ones that people have mentioned here).
Learn how to carry your belongings safely.
And how to run quickly