Fairly early on, choose a thesis topic or problem you're interested in, and pursue it. Don't get side-tracked into something just because it's what the department or your advisor is working on. In fact, I'd go so far to say you should pick your thesis topic before you apply. Use that to help you decide where to apply. If you don't have a thesis topic you're committed to, you're not ready for grad school; and should stay out.

This may actually depend on the field. In PhD programs in linguistics in the US, you are usually expected to pick you thesis topic only in the third of five years, and you actually have to work on other topics, too. The programs are structured in a way that you wouldn't even have the time to focus on your thesis topic right from the beginning.