Can I say: Joe is a better golfer than me. or do I have to say: Joe is a better golfer than I am.
Both. If there's a verb, use "I", and if there isn't a verb, use "me":
Verb: Max is taller than I am. ('am' is a verb)
No Verb: Max is taller than me.
Please Note, speakers who subscribe to traditional grammar rules would use "I":
EX: Max is taller than I.
All the best,