In general people fly to Lanzarote at Arrecife (ACE) Airport.
The Airport is siruated between Arrecife (the Capital) and Puerto del Carmen.
If you are hiring a Car, you will pick it up at the airport. You can also travel to your final destination by Taxi ( cost approx 12 Euros to Puerto del Carmen, 15 Euros to Costa Teguise and 30 Euros to Playa Blanca)

You can also travel to Lanzarote from Spain by boat, although the service is quite limited