A good way to get to know the historical center is by taking a walking tour. When your feet get tired, hop on the Turibus to enjoy the sights from a different perspective. And when hunger strikes, head to the Hosteria Santo Domingo for a great meal.
Mexico City is an excellent destination for travelers on a tight budget. There are lots of activities you can do for free, see Mexico's Top Free Things To Do, and it's quite easy to find a decent, cheap hotels. Here are a few: Cheap Mexico City Hotels.
Mexico City gets a bad rap; it's not as dangerous as many people think. However, like in any big city, you should be sure to take general safety precautions against theft.