Best beaches are the Papgayo beaches at Playa Blanca. Lanzarote isn't big on nightlife, but Puerto del Carmen is the best resort for it. By the way, if you ask for a nightclub you will be asking for a brothel. Discotecas are what you need.

Taking both your requirements into consideration you should go to Puerto Del Carmen, accommodation as near the old town as possible. The beaches are perfectly OK although the main one Playa Dorada is very open and can be a bit windswept at times, and in the summer cooking hot with no shelter other than pay for a sunbed/brolly, so take your own. The beach at Papagayo, Playa Blanca, is certainly nice, part of it is a nude beach, if that's your thing. Drawback is it's at the extreme end of the resort and fairly isolated. So either a car or a long walk. You won't be exactly falling over the nightlife, clubs, discos, in Puerto Del Carmen as there's not lots of it. Lots of bars with sport and music.