Any tips on visiting these 2 must-see items on our agenda? We will be staying near the main square in Krakow. Should we book a tour ahead of time? We are told we might want to get tickets ahead of time, but can we just take public transport to the mines?
From the area you are staying in you can easily walk to the castle. Only part of it requires a ticket. We didn't get one. Salt mines. Your hotel will be able to tell you how to arrange to get there. We found an old farmer soliciting rides to the mines. He was sitting in front of the old train station. The Jagiollonian university is close by and is worth a visit. The Oskar Schinler Museum is a long walk and it would be easier to get there to take a pedal taxi or a bus. You can do this all on your own. There is an excellent Tourist Information office on one side of the Cloth Merchants Hall which sits in the middle of the square. That's the first place to stop for help.
If you can, go to the castle right when it opens. It gets packed later on. Ditto on "Lady with an Ermine" comment.
Ditto what Monte said. Go early or late to the castle area - best time to see the cathedral was in the early morning (IMHO) If you have the opportunity, buy a $3 ticket to see the daVinci painting "Lady with an Ermine" - it's a rare chance to see a single painting in a very intimate setting (it's a little room with just the painting). It was the best money I spent in Krakow. (see http://www.wawel.krakow.pl/en/index.php?op=13)
If you would like to see what the Rynek Glowny (central square) looks like there is a webcam: http://krakow.pl/multimedialny_krakow/3222,artyku. Someone posted this more than a year ago and I look at it nearly every day. Remember the time change is nine hours different, depending on the time of year. The train staion square is called Krakow Glowny.
I'll address the Salt Mine part of your question (agreeing with others on the castle and other sites). We had a rental car when we went to the Salt Mine. We got there right when they opened and took the next available English language tour. By the time we emerged, there were long lines for all the tours and a wait of something on the order of two hours. This was in August, which is peak season, but it was still totally nuts. If you go on your own, get there as early as you can to avoid the wait. If you go with a package company, see if you can determine in advance if they have a specific tour booked. I enjoyed the mines, but I'm not sure they would have been worth the long wait.
I visited the salt mine with a half-day tour, arranged through my accommodation. It's definitely worth a visit! It was an easy way to get there and I enjoyed the tour. The mine was very busy, but we just breezed through. I'd do the tour again, as I found the convenience and commentary worth it. I didn't feel rushed and wouldn't really have spent more time there. The big plus for me was that my knee could not handle the long staircase down, so the tour guide arranged for me to take a special lift down. Individual visitors would not get this access, as I understood it. By all means though, do the rest on your own.
I loved the mines, and took a half-day bus tour arranged by the hotel. When I realized how closed the mines actually were to the city-center I felt like an idiot:) The mines are in the suburbs and connected by a freeway, so if I were to do it all over again I would take a taxi.
Yes you can take public transportation to the mines. There is a local train from Krakow which stops in the town where the salt mine is located. The salt mine entrance is a short walk from the train station. As I recall the train tickets were very inexpensive. The train itself had certainly seen better days but is got us there and back - just part of the adventure. We were there in May and did have a short wait for our tour of the mine as there were a lot of school groups there at the time. The mine experience was very interesting and fun. I would highly recommend it. We absolutely loved Krakow - enjoy your visit there.