I'd like to know if it's legal to play covers in public

Well, I guess it depends on the laws in your particular country, but I can't see any reason why it wouldn't be ok. So long as you don't record the gig and try to sell it on CD or something, with other bands songs on it.
 
someone explained to me that its ok as long as the club has paid yearly fees to ascap and the like... im pretty sure its the clubs responsibility.