I'd like to visit Wieliczka salt mines and the Auschwitz KZ. Is it possible to visit both places the same day? How much time should I stay in each place? What trains/buses have to take from central Krakow? Also, I'm wondering if it will be cheaper and more cost-effective to join to an organized group or do it on my own?
It's possible to do both in one day, but I wouldn't recommend it - it's a lot to take in, and makes for a very long day. There's so much to see in Krakow old town, that it's worth spreading them out and spending your down time on a city tour or just walking around trying random foods in old restaurants or seeing the performers in the main square, or just sitting in the parks.
What I did can be read on my blog, so I'll include the links:
They are run by Cracow City Tours, and were pretty reasonable.
Now, the reason I suggest a tour. For the mines, you'll find that as an individual it's possible to get out there (see Wikitravel for bus numbers and directions), but then you'll have to find a tourist group to latch onto, as you can't explore them on your own. Also possible, you just have to ask around.
For the Auschwitz and Birkenau tour, it's one of the most depressing (but important) tours to do, and personally, I found it better to make some new friends on the tour to have someone to talk to about it. We had an Israeli guy with us who'd had relatives at the site, and that was really interesting. But it's quite a moving place, and I'm glad I didn't do it alone, to put it simply.
Also really recommend Greg and Tom's Hostel, by the way - 3 min walk from the train station!
It's not easy and not recommended to do both in one day!
The best is either to have the private car or to choose the rush tour starting very early morning:
The tour of Wieliczka Salt Mine takes at least 3 hours. The location of the mine is half an hour drive by car from the center of Krakow. The town of Oswiecim (in German: Auschwitz) is about at least 1 hour and 15 minutes drive - better to say 1,5 h drive by car from Wieliczka (the highway is distant). And you need there at least 2hours to see the most important parts of Auschwitz KZ.
You should add an extra 1h for traffic jams (very common in Krakow) and it is: 3,5-4,5 hours of drive + circa about 5-6 hours for both tours (it's not easy to shorten them, especially in Wieliczka Salt Mine where is almost obligatory to have the reservation for the entrance). So it can be a tiring 10 hours without lunch and stops. Let's say 11-12 hours.