Get a good razor scraper, one with a thumbscrew that tightens up the blade like this:

http://paceperformance.com/images/F30824404.jpg
Don't **** around with one of those POS retractable blade scrapers, it will do nothing but piss you off.

Get everything you can with a razor blade scraper - without getting into the tub finish. Get the rest with a chemical. Try mineral spirits, goof off, gel stripper...

Just turn the water on really cold for a few minutes. Your caulk won't fall off, but it will shrivel up to the point where you can barely even notice it's there.

Tissue?