Can I ask why you are looking to stay in Oswiecim, and not Krakow? I've only ever visited the camps, but my recollection is that there's not much to do in the town besides the touring the camps - which is draining. I guess I'm just trying to imagine your first few days on what sounds like an incredible trip. You'll arrive in Prague, jet-lagged, and sleep - then wake up early the next morning to catch a train or private car to take you to Oswiecim, where you'll arrive - midafternoon? I did a quick search on MyDayTrips and it looks like a 9am departure gets you into town around 1:30pm. You'll spend the afternoon and evening there, then tour Auschwitz the next morning/afternoon, and then - go back to your hotel room? Then head down to Budapest, which is at least 8h30m by train. It seems like a lot of work to get to Auschwitz, and almost a waste to be so close to Krakow and not see it. Two afternoons and evenings can be well-spent there, and especially after a day at Auschwitz, it might be good to escape back to the Rynek, people watch, and explore the Cloth Hall. Just something to consider. (Another thing to consider - flying into Krakow.)
As for your actual question - I would take the train from Prague to Oswiecim. There's a direct that leaves around 10am and gets in around 1530. My very superficial search on MyDayTrips said the drive from Prague to Oswiecim was about $250, and I can't imagine two train tickets would cost nearly that much. That being said, I can't prove that, because when I typed in your route on rozklad, I was told I could buy the tickets at the station but was given no price. Perhaps someone else will have better luck finding the price. As for the trip to Budapest, check out airline prices leaving from Krakow or even Katowice. On skyscanner there are flights listed in September for as low as $66 for a direct, one hour flight from Krakow to Budapest.