The buses in Bad Kissingen drive only from Monday to Saturday noon. It is for free if you show the driver your "guest card" (which you recieve for paying the spa tax at your hotel). I recommend to visit Bad Kissingen by car as taking the bus from one end to the other end of the small city can take an eternity.
There is another bus called "Bäderandbus" which connects the cities of Bad Brückenau, Bad Bocklet, Bad Kissingen, Bad Neustadt and Bad Königshofen between May and October. All information you can find on the websites.
Leave the car behind and take the train! Public transportation is so easy. The train station lies within a 10-15 minute of the Kur house.
The train station is also a good place to park for just 1 Euro. You can park you car here all day. Post your ticket stud on the dash visible from outside on the driver's side of the car.
This little blue shuttle runs around the city. Easy to spot. For healthy people, most of the center of town can be seen by foot.