Hi Tez,
The HIPERDINO supermarket is right opposite the airport. It is a very large hypermarket with parking, selling everything you could think of, both food and non-food, all the locals shop there. It has a large deli counter selling meats & Cheeses, a butchery and a fresh fish counter, also sells lots of branded UK items such as PG Tips. When we go to Lanzarote we always pick up our hire car at the airport and head straight there to get our mainstay items and drinks and then just top up with bread / rolls milk etc at the local shops once in resort. The hypermarket is a 5 minute drive out of Puerto del Carmen, on a service round alongside the main dual carriageway. You can see it as you leave the airport. Once in resport itself the SPAR shops are fairly good value. Hope this helps