We spent four days walking around Krakow on our trip there in 2008 and that was plenty of time to see all the main attractions.
We had a rental car and drove to Auschwitz Concentration Camps . There's a parking lot there and a bus that will take you to Birkenau which is included in your visit ticket and close by. You need most of a day to see them both. You don't need to book a tour.
While in Krakow , definitely visit the Jewish district of Kazimierz . They have several synogogues, a beautiful old Jewish cemetery, and a few really good restaurants. We went to one that only served pierogis - wonderful.
The Wawel Royal Castle and adjoining church are both stunning and worth the time to do the tours. You can spend quite a bit of time here. Walk around, see the Dragon Cave, and then meander the tiny streets towards the Main Square (known as Rynek Główny ). There's a beautiful old market building in the center filled with vender stalls selling leather items, amber, dolls, and many other things. It's crowded and busy, but fun. The church is gorgeous. Do go inside. Here you can also catch horse carriage rides through the town.
I never used our car while in Krakow. It's an easy city to walk, so we walked everywhere and really enjoyed our time. We went in October, so it was cold. Shopping is great.
While you're there, try Bigos. It's a traditional "hunter's stew" made with meat and cabbage and is delicious, but different at every place I tried it.
Yes, you can rent a car easily enough. But I'd still take the bus or one of the new trains. Much easier and less complicated than driving. It's really the middle of nowhere. The experience can also be quite emotional. I wouldn't want the stress of driving added to it.
You should be aware that in the Winter, the Birkenau camp is closed. Check before you go.
I'd definitely book a guide before you go. Not many tour guides hanging around in the middle of winter hoping tourists show up. The camps are VERY exposed to the elements. Dress warm!
Krakow is a very interesting place. There's always something going on but you may have to hunt to find it. The majority of places in the city center are underground. Best to make friends with local and have them show you around. 'Couchsurfing' is a good place for that. You can arrange to meet people for a coffee, etc. no need to actually couchsurf.
On my first trip to Krakow, my 5 day stay ended up being extended over and over again to 12 days. I never ran out of things to do in the town. In addition to meeting my future wife, I was amazed by the architecture. What a shame that the use of coal leaves the buildings dark grey!
They have daily trips from Krakow to Oswiecim (Auschwitz, Oswiecim in Polish). If you take a trip by bus, it may get complicated. Auschwitz has guided tours or you rent a little portable guide tablet that you can listen to. But if you choose to walk around the camp by yourself, everything is well organized and explained to you in writing.