Matera is a fairly big city. One of its quarters is the old city. Just outside the old city are the Sassi - cave houses dug into the limestone rocks. The sassi are located on both sides of a valley. On the north side the Sasso Barisano have been partially commercially developed as offices and housing. The Sasso Caveoso, on the south side, are the most historically interesting.
The Women's Fiction Festival is held late every September in the Sassi District, and attracts an international contingent of writers, agents and editors. FMI http://www.womensfictionfestival.com
There are many restaurants in the old city as well as outside.
There are common hotels in Matera. But sleeping in the caves is a completely different experience. There are hotels like [http://www.hotelsantangelosassi.it Hotel Sant'Angelo] made up of 16 historical dwellings, each with an average surface area of 50 square meters that correspond with other cave dwellings whose interiors have been reproduced and divided into various dwellings with living rooms. All rooms are furnished with: private bath, telephone, satellite television, air conditioning, a safe and mini-bar. But consider that you can enjoy the place staying in one of the many B&B's in the old town. Like [http://www.sassicasastella.it B&B Casastella] (Near to St. Barbara rocky church). The property is in part cut into the rock, with the remaining structure built on four levels with characteristic barrel-vaulted ceilings, restored from characteristic peasant dwellings such as the granary, stable, barn and wine cellar.
There is a tourist office at Viti Marco and one at the old city. The Viti Marco one has English-speaking staff which is very helpful and will give you a map and a booklet on Basilicata