Buses leave frequently from Katowice's main bus station (ul. Skargi 1) every day starting at 8:00. A one way ticket costs about 9-20zł with the journey taking 60-70mins (prices and times depend on the company). The buses make several stops along the way, and most weekday will drop you off at the Oświęcim train station (ul. Powstańców Śląskich 22), which is 1.5km away. From there hop on local buses 24, 25, 26, 27, 28 or 29 to make the short trip to the museum; you can purchase the 2.70zł fare from one of the nearby kiosks.
Auschwitz I and Auschwitz II-Birkenau are located 3km apart (with the train station strategically between them). Between Nov-Feb there is a free museum bus which shuttles visitors between the two parts of the camp every 30 minutes during weekdays from 10:00 (every 10 mins at weekends), with the last bus being at 16:10 (check upon arrival for schedule). Alternatively, catch a cab for a rich 15zł.
To get back to Katowice, waiting minibus taxis run by Malarek Tour (+48 605 31 50 77) can be found outside either camp - a group of eight would pay about 30zł/person. Otherwise you need to catch a bus from the Oświęcim train station. Again, local buses 19, 24, 25, 26, 27, 28 or 29 make the trip from Auschwitz I to the train station for 3zł. Buses from there back to Katowice are frequent, but the last one leaves at 16:48 Mon-Sun.