On your own, you can go from Málaga directly to the caves of Nerja by bus. You can search for the times, prices and buy your tickets here by searching for a trip from Málaga to Nerja (Cuevas). You’ll need to take the bus from the bus station just outside María Zambrano.
Once in the caves, you’ll appreciate it if you have bought the tickets in advance. You can do that here. Every Monday from 9 am to 1 pm the entrance to the cave is free but you have to still make the reservation.
As regards to the organized tour of the caves, you do not have to sign up for the tour that starts from Malaga. That said, the caves are very big and you might want to take a guided tour so that you don’t skip any part that might be important. The tours also tell interesting stories about the caves and explain the history of how they were discovered etcetera. However, the regular entry comes with an audio guide that will probably explain the essentials, so if you want to save money, an audio guide would be a good option.
The Caves festival is held in the month July. You can get details at the official site.