It is incredible easy to move around the island with a car. Rental cars are extremely affordable compared to many other regions of Europe. It is worth to take a car for at least 2 or 3 days now that you know how big is Gran Canaria.
If not, you can still get around easily, although not everywhere. If you are in Las Palmas, you have the best options. You can reach the south of the island (Maspalomas) with a bus ride within an hour. Although be careful: look for the direct lines. If you take one of the indirect lines you will be stopping in every single village between Las Palmas and Maspalomas. A 45 minutes ride may become a 2 hours ride. It is always worth to wait for the next direct bus. From Las Palmas you can easily reach places likes Agaete, Gáldar, Moya or Arucas in the north, and Teror or San Mateo in the center of the island. As well as any other location in the south: always keep in mind that depending on the place you might have to take an indirect bus with many stops.
While you can visit many places in the same day with a car you can still visit many of these places by bus but in different days. The peaks of the island (Roque Nublo, Pico de Las Nieves) are worth visiting and they are are not so easy to reach with public transport. There is actually one bus that only drives from Maspalomas, and only a few times a day to a place close to Roque Nublo. If you want to see the peaks of the island, you should rent a car or book an excursión.