"I am sick today."
I think so, I was taught that the time expression can always be at the beginning of a Chinese sentence, and for me 我今天生病了 sounds like it has an emphasis on 今天.
what is the difference between 有病了 and 生病了 here? Do they both mean to be unwell or sick?
有病 sounds a little bit more serious to me than 生病, but that might be completely wrong (I'm not a native speaker). However "有病" can also mean "to have a mental problem" or just note somebody's crazy behaviour.