care to (do something)
1. To be interested in doing a particular action or activity. Often used in the negative to mean the opposite. No thanks, I don't care to jog today—my legs still hurt from yesterday's trip to the gym.
2. Would you like to (do a particular action or activity)? I'm going out for a jog—care to join me?
care to do something
to want to do something; to be willing to do something. I really don't care to see that movie. Would you care to go out for dinner?