Intercity Bus Lanzarote operates a fairly comprehensive network of buses around the island, though if you're hopping between small towns you'll need to plan carefully as services are infrequent.
Practical Tip: Bono Bus
If you are planning on making a few trips, invest in a Bono Bus card, which can save you up to 30% off the fare. The cards cost €10, €20 or €30 (depending on how many trips you plan on taking) and are purchased at bus stations. Tell the driver your destination and the amount will be deducted from your card via a stamping machine.
Driving on Lanzarote is the best way to see the island. You'll find plenty of car rental agencies at the airport, in Arrecife and in the main tourist resorts.