Note: The advice below is specific to the Neural Networks Group. Please add more sections for generic advice, or specific advice for other groups.
Typically you get started with research by doing a project for a class such as cs394n (the graduate class on Neural Networks). In some case, the cs396 project might turn out to be a good starting point as well. It is a good idea to start with something small and manageable, and expand it as you make progress. If you try to take too big a bite in the beginning, you may not get conclusive results, and it is hard to figure out how to continue. You are encouraged to come up with your own ideas for projects, but I also have a large list of project topics (in the cs file system) to suggest. Many of these are offshoots of existing work in the group. Building on other people's work is always a good idea to get started, because there's already a lot of expertise around and you can hit the ground running.
After your initial success in cs394n or cs396, we usually continue the research with a number of cs395 follow-ups. These are independent study courses where the idea is simply to get academic credit for working on research. In the beginning, you usually take one cs395 along with your breadth or depth classes; when you are closer to your proposal, you may take three cs395 classes and concentrate only on research. These classes are pass/fail, and you get a pass if you make reasonable progress. We set up goals in the beginning of the semester (see my research plans file in the cs file system) and revise them as we go along. If you have never taken a cs395 or cs394n with me before, I ask you to write up a paper on your research at the end of the semester. If you get significant results, we might even turn that paper into a conference submission.
It is perfectly ok to explore different topics while you are doing cs395 courses, and even different advisors. The idea is to get an idea what research is like in this area, and which topics would be fun to turn into a dissertation. There is no particular hurry to decide on a topic during your first two years here. Explore and have fun. After that, you should have a pretty good idea of what you would like to do, and it should not be too hard to narrow it down to a dissertation topic.
