For roller coasters, they have two different shops. They have the wooden roller coaster shop where the trains for Racer, The Beast, Mystic Timbers (after Winterfest), and Woodstock Express are taken to. And for all the steel roller coasters there is also a shop. During the off season, the trains on all the roller coasters in both shops are disassembled completely, cleaned, inspected, and then reassembled for the next season. For flat rides, as far as I am aware of, most if not all the maintenance is done right there at the ride. I know that Monster is virtually disassembled at the end of every season and I believe the logs in the log flume are taken out and stored in the picnic grove as well as other parts to other rides. The buildings these are most likely maintained in are the buildings on the north side of Racer outside of Area 72. And the steam engines on the KIMVRR are maintained in the engine house on the railroad. The buildings at the front end of the park are office buildings as well as warehouse space for other departments of the park.
Fun fact: If I remember correctly, I saw somewhere where Kings Island spends a good $100,000 on just wheels for the roller coasters.