If there's no "self" then why should I care about the future lives and nirvana?
Whether you believe in a 'self' or not, suffering is very real. You experience it on a daily basis. It is this suffering that makes beings look for an end to it. There's only one permanent end to suffering and that is Nibbana. That is why you should care about striving for it. Craving is what prevents you from accomplishing this. Ignorance is what causes craving to arise. The belief of a self is just one form of ignorance that you have to get rid of in order cut off craving.

Ignorance means not seeing things as they really are. So you cannot get rid of it just by changing your opinion or reasoning it out. You need to start seeing things as they really are.