Yes, I agree that traveling all the way to Poland to--effectively--see very little besides Auschwitz seems misguided. Krakow is a lovely city.
For Auschwitz you'll need to take a bus. Public buses make the trip fairly often. You definitely should buy your Auschwitz admission ticket ahead of time. At the time of my visit in 2018, advance tickets were sold out for the next 8 days. Going without a ticket means waiting in a very long ticket line at the camp/memorial. The line stretches across an open area with no shade or protection from the weather. And there's no telling for what time (if at all) you'd be able to get a ticket if you just showed up.
Auschwitz tickets are time-specific, so you must make sure you get there on time. I was a bit concerned that the bus might sell out (don't know whether it ever does), so I walked over to the Krakow bus station (beyond the train station) and bought my ticket the day before. There may be a way to do that online, but I want to be sure I knew how to find the bus station; the location is a bit odd.
The public bus was fine. There's no real advantage to signing up for a commercial tour unless you will get useful commentary on the way to and from the camp--or unless you find no tickets available for your date from the official website. All tours inside the camp are conducted by camp personnel, no matter whose bus you arrive on.
One other thing: Poland is a comparatively inexpensive destination, but you will pay tourist-site prices for anything you purchase (water, soda, etc.) at the camp.