We went to the Callista Catacombs on the public bus (660) to Appia Antica. Unfortunately we got off the second bus two stops early; we walked at least a couple kilometers.
The tour of the Callista catacombs was very nice. An Australian priest named Father Owen led us. He is a member of an order dedicated to preserving and caring for the catacombs. He gave us a quick overview of catacomb history and then of the symbology. Callista is the oldest catacombs, and has 13 popes, many martyrs (including St. Cecilia), and four levels underground (note that we found out at Callista, that one of the levels has completely collapsed, the one closest to the top.)
The catacombs are named that way because the word means something about the garden (double check this). It was named this because the catacombs of Saint Sebastian fit this description. Most people in Rome in that day were buried in similar structures (although the pagan Romans were cremated first). Often families would be buried in subterranean vaults. The difference with the Christians was that they wanted to be buried together as a community, hence the vast catacombs, with tens of thousands of people interred. They dug out the Tufa, a volcanic rock that was fairly easy to dig, wrapped bodies and anointed them with oil, then closed up the shelves with marble slabs (or stone/brick in some cases) which they mortared in.
There is a certain amount of art in the catacombs, from sculpture (the good shepherd sculpture, now in the Vatican museum is a good example) to painting. All incorporate positive Christian symbols like the good shepherd, the holy spirit, the miracle of the loaves and the fishes. The symbology, no surprise here, is very oriented toward the resurrection.
If you are a interested in early Christian history, the Catacombs are a must, but most any catacombs will do. If you are particularly interested in the Art, these are the ones to go to, since they had the best art of the three Catacombs we visited.
Web site for the catacombs is: http://www.catacombe.roma.it/