We have been to Matera on 2 occasions. The first time (2021) we simply walked through the old town and thoroughly enjoyed it. On this trip, (Oct 2022) we thought we'd hire a guide for a more personal experience. The lesser expensive tour twice cancelled on us at the last minute, directing us to the much more expensive tours, no thanks. We approached the local tour shop who tried to scam us with fees as well. They also said that Americans were not allowed to access the caves.
Long story short, read up on the history of Matera and walk it on your own. There are no fees for entering the old town, and no fees for hiking up into the caves. It is a wonderful experience, and it's absolutely free! There are many bars and restaurants in the old town and it's easy to get lost wandering through the streets...and great fun!