I broke my toe two weeks before my trip to Central Europe last year so I brought a foldable luggage roller for my large backpack. The sidewalks from the train station to Old Town were smooth and easy for rolling my backpack to our apartment lodging, located near St. Mary's Basilica and the main market square, where my family and I were staying. There are some cobblestones in the Old Town but if the wheels on your suitcase are in good shape, you won't have any trouble going over them. My family and I really loved sleeping next to the square since there's so much to see and getting up early and walking into the main market square with only a few people about in the morning was magical! It was nice to walk around the square after dark too to see the buildings lit up. Surprising, since there are quite a few people walking about. The square's restaurants and outdoor eating areas are crowded with late night diners so you feel quite safe. It really is beautiful to view the churches and Old Town areas at night. If you are walking outside the main square at night, like through the park or along quiet streets on the outside of the historical area, then I might feel nervous or unsafe. Overall though, the people are so friendly and helpful. I have never seen so many families with young children on my travels as I saw in Krakow...and parents took them out late in the evening especially on the weekend nights. I think you might miss out on the historical beauty if you do not stay in the Old Town. I felt it was more modern and actually crowded near the train station since to exit the train station, you walk through a very modern mall. Rick Steves never seems to go wrong with his hotel recommendations and he lists quite a few for Krakow in his Central Europe book. You might want to look into them especially if you are arriving to Krakow and departing Krakow in daylight hours. But...there's always Uber.
On our trip to Krakow last year, we took the train from Krakow to Oswiecim to visit Auschwitz. From Oswiecim's train station, we took the bus, which runs quite frequently, to Auschwitz I. The official Auschwitz web site is packed with information and detailed instructions on how to travel by various methods to Auschwitz I and even provides the bus schedule from Oswiecim's train station. For our return trip back to Krakow, we took the bus from Birkenau to Auschwitz I, transferred to another bus which then took us back to the train station for our train trip back to Krakow. Make certain to reserve your day/ time slot/ language for the tour at Auschwitz.
While in Krakow, we mostly walked and sometimes used the tram. To make the best use of our time since we only had a few days in Krakow, we took an Uber to the Divine Mercy Sanctuary and walked from there to the Sanctuary of St. John Paul II. Our Uber driver was a friendly, young college student and the cost was so cheap. We paid by cash. We also used Uber to get back to our apartment. Uber drivers are not allowed to drive in the Old Town so our driver took us to the closest street to our apartment in the Old Town.
You will have a wonderful trip no matter where you decide to stay. Krakow is so beautiful and full of history!