Here's a list:
- collaborate on a research project with a friend or someone that you can meet on the Internet.
- contribute to an open-source Bioinformatics project, like BioPerl or BioRuby.
- read a journal that discusses an algorithm that you find interesting, and try to replicate their results. If you're not at a University, some of the good open-access journals include the PLOS series of journals, old issues of PNAS, and Bioinformatics. Keep in mind that in computer science terms, many bioinformatic algorithms are a little bit boring.
- take your favorite problem from your experience in molecular biology, and do research on it at some of the large databases, like the EMBL or the NCBI. There's a great deal of data out there.
In general it's hard to know what questions to ask when you come to a new field, so if you can find a mentor/collaborator that would be optimal. Barring that, read a lot and pursue what you find interesting. Good luck!
Edit: Since you asked for reading requests, I would recommend this book:
Biological Sequence Analysis
This book is suitable for upper-level undergraduates and graduate students with moderate experience in computation. It's a good reference, and does a very good job explaining the motivating ideas behind some of the more important algorithms in Bioinformatics.