For postdoc applicants: I currently have no funding for postdocs.
For Ph.D. applicants: A word of warning: professors get a lot of emails asking about research positions. Most of these are mass-emailed form letters and are ignored. Even if yours is not, it's a good idea to understand how to make it stand out so it doesn't look like a mass-emailed form letter.
I don't expect a prospective student to have read all of my papers or solved open problems from them. I do, however, expect a prospective student to know
But otherwise, it's a good idea to talk about a creative project you've worked on, preferably outside of class (or that started in class but continued far beyond the expectations for a good grade), something that you are proud of, and how you think it demonstrates that you might have what it takes to do research in theoretical computer science. Ideally, this will be a previous research project in theoretical computer science, but not necessarily.
I recommend reading the Andy Drucker's advice for some more ideas.
For UC-Davis undergraduate students interested in research: The advice to Ph.D. applicants above applies. Additionally, since you're already at UC-Davis, note that I occasionally teach a course on my research, ECS 289A: Theory of Molecular Computation. Taking this class is a good way to learn about foundational research in my field and can be the start of a research project. For instance, a student once got so interested in Optional Problem 5 in this homework, solving the homework problem, then improving the solution, generalizing it, wondering how far it could be extended, until he eventually turned it into this paper, which was published in PODC, the leading conference on distributed computing. It all starts with following your curiosity as far as it takes you!
I also frequently teach ECS 220, graduate Theory of Computation. It is less directly related to my research, but it's still a good course to take to see what sort of problems I find interesting (and to demonstrate to me your problem-solving skills).
For undergraduates, timing is an important issue: most research projects take a while, and a single quarter or summer is not enough time to complete it. Rather, it's enough time to start it, and you might spend the whole next year actually getting the results and writing a paper. So it's best to look into this before your final year.