"Thankyou" (informal or formal)
"Thankyou very much"
"Thanks a lot" (casual - if you say it without emotion it'll often be mistaken for sarcasm)
I've heard say "thanking you", but this always sounds awkward. I suspect this is more common in the UK or US.
"Many thanks" (it can sound a little archaic)
"Thankyou/Thanks for ...." (You can say "Thanks for the message", "Thanks for doing that", "Thanks for the gift" etc.)
I'm struggling to think of ways to thank someone without saying 'thank'. Here's one:
"I appreciate it/that" (after someone's done something for you or offered to - you can combine it with others by say something like "Thankyou, I appreciate that" or "I appreciate that, thanks")