Best city in Germany

Susan, For visiting Neuschwanstein and Hohenschwangau, Munich is about the most convenient home base. It's an easy trip to Füssen by train, and then a short Bus ride from the station. You can visit the Castles on your own, but you'll have to book the time-specific tour yourself (don't be late!). You could also take a day tour with Radius Tours, and they'll take care of all the transportation and reservation bookings. Note that you'll have to walk uphill to Neuschwanstein to enter the Castle. There's also a horse drawn carriage for those that don't want to walk, but I'm not sure what the cost is for that. I've never been to Ludwisburg, but Munich also appears to be a reasonably good location for that. Were you planning to just take a day trip there? The train will be about 2.5 hours from Munich each way, with one change in Stuttgart. Be sure to check the operating hours for the sites you want to visit. Happy travels!

By "Sleeping Beauty castle", do you mean Neuschwanstein? There's another castle in Germany that is specifically known as Sleeping Beauty's castle. Those two destinations aren't really that convenient to each other from the same base, but if you had to visit them from one location, I would say perhaps Stuttgart, maybe Ulm.

"There's also a horse drawn carriage for those that don't want to walk, but I'm not sure what the cost is for that." I don't remember the exact cost, but it's minimal. Only a few euro.

The horse carriage costs 6€ up, 9€ round trip, per person. The carriages leave from in front of the Hotel Müller, just up the hill from the ticket kiosk. Munich would be the best for the castles; Stuttgart would be better for Ludwigsburg. Ulm or Augsburg might be a good base for both.