Could cancer make it painful for me to urinate, or is that more likely to be an std?

Hmmm. Could be many things....Are you having burning on urination? I would say there are many other reasons for pain on urination, that are more common than cancer - urinary tract infections (including stds), trauma.... Would recommend you see your pcp and get a urinalysis, possibly a culture, and physical exams to figure this out.
Maybe. Either way, this is a significant issue that needs to be evaluated asap. Call you doc and get an appointment and exam.