Aerating is indeed very good for your lawn. This should be done in early spring, after the first lawn mowing. In combination with sowing and fertilising you should get back to a beautiful lawn.
Regarding the moles – they are a protected species. So you are not allowed to banish them off your lawn. But there are kind ways to tell the mole to leave. Moles have a very delicate sense of smell. So in the specialist shop you will find little lemon-scented pieces of wood. Just put them into the moles tunnel and it should leave. After the mole hase left, make sure to destroy the tunnels. Otherwise the root vole could take over - and this is worse, for sure.