There is one  airports on Lanzarote. Near Recife..  The airport operator is Aena.  Information is available from their web site.  Click here for on line  flights
 

If you need aTaxi, agree on a price before the start.  Else use car hire.

An alternative is to book transfer together with your hotel. This service is offered by www.alpharooms.com . An other possibility is www.resorthoppa.com    A2Btransfers.com
or shuttledirect.com

If you are planning to use public transport, the trick is to ask the tourist office at your destination. To find their address & Tel. or e/mail, use the Lanzarote tourist board web http://www.turismolanzarote.com/turismo.
 

The bus transport system has frequent connections to Teguise and Puerto del Carmen. Less frequent to Playa Blanca. A complete time table can be found with Lanzarote Gazettelive.com