Go to the small town of Teguise which is up in the mountains confusingly not in the resort of Costa Teguise! It's really lovely with small boutiques, tapas bars, old buildings & amazing views, on a Friday there's a Very busy market.
My sister and I always hire a car. We love Lanzarote.
The Caesar Manrique house & museum are worth a visit, also Cave d' Jardin which is beautiful large cave with an inside lake & outside pool; the Cactus Gardens, & the caves which can be a bit claustrophobic though. Also Omar Sharif's old home is interesting to visit.
The best beaches are the lagoon at Playa Caleton; Playa Pocillos & Playa Matagorda near Playa del Carmen, and the beaches of Pozo & Papagoyo in the nature reserve near Playa Blanca.
The beach at Farmera is too dangerous for swimming but it's good to go & watch the surfers & kite surfers.
Near Playa del Carmen is the Marina of Playa Caleta which has some great shops, restaurants & bars plus a market.
Also there's the Biosphera shopping centre slightly out of town with a large Zara etc.
It's definitely worth visiting the Timanfaya National park with all the volcanoes & lava fields, & the seeing the green volcanic lake on a nearby beach.
There are lots of quaint towns to visit as well plus you can take boat trips.
By the way I've never spent time in Costa Teguise or the Playa Blanca resort.