I live and work in CA for an employer in San Francisco, but I travel often to FL and work there remotely for a total of perhaps 2-3 months a year. Am I supposed to pay state income taxes to CA during those months even though I am earning income and working in FL?
The good news is, Florida doesn't have a state income tax, so you owe Florida nothing.
The bad news is, California's tax law says that you're taxed on all income earned from the State while you're a resident or part-time resident (which you appear to qualify as)
TL;DR: You owe Cali taxes 365 days a year on all income generated from a source in the State of California (IE your job).
If you were less lucky, you might owe taxes in two states at the same time as a part-time resident of one and an earner in another.