A Web programmer interested in software testing? Cool! Let me start out by saying that if you are exploring a personal interest as a result of the course you took, I suspect that you are embarking on an enjoyable journey. Testing is, or at least can be, a rich, exciting and rewarding field, but most folks don't give it much of a chance unless they are lucky enough to have a great first experience or a good teacher/mentor.
The number one way to improve your skills is to practice. Test everything. As a programmer, I suspect you sit at a keyboard all day, giving you plenty of opportunities to test things. Volunteer some time to test for an open source project. That sort of thing.
Next is to learn everything you can about everything you can think of. Not just technical stuff, not just testing skills and techniques, not just management practices for software development shops. For example, learn about human psychology (specifically how humans interact with machines, what they expect cognitively, etc). Learn about presentation of graphical information, learn about statistics, about discrete mathematics, general systems thinking, operations research and color combinations that are most likely to create problems with users who experience varying degrees of color blindness. (That's just a short list of examples that came to me as I am writing this on a plane at 2 in the morning.)
Finally, there are three books that I believe are absolutely "required reading" for anyone who is serious about software testing:
- Testing Computer Software (2nd Edition; Kaner, Falk, Nguyen)
- Lessons Learned in Software Testing (Kaner, Bach, Pettichord)
- General Systems Thinking (Weinberg)