Upvote:2
There is a list of castles in Romania on Wikipedia, you can figure your favourite from this list. Reaching some of them within a day by public transport is possible, for example:
Iulis Hasdeu castle, in Campina, is 1h15 from Bucharest by train,
Făgăraș castle may be a bit too far off, reachable after 4 hours of train
and the others, including Corvin castle, seem to be too far from Bucharest to be reached within a day. Whether a rental car is more comfortable is your choice, depending on your preferences (cost or comfort).