Just to clarify, the correct spelling of the town name is Oswiecim. You'll probably need the correct spelling if you research train schedules, etc.

If you arrive in Oswiecim by train, it's only about 20-30 PLN to go to Museum Auschwitz by taxi. Also, you can take a taxi from Auschwitz to Birkenau, and usually the taxi will wait outside so that after you're done with Birkenau, they'll take you to the train station or wherever. Handy if you're pressed for time.

If you are not going back to Krakow afterwards, Katowice is about an hour away and a good point to catch a train, especially onward to Warsaw, Budapest, Vienna, or Prague.