You can look up trains at www.bahn.de (you may need to use the Romanian spellings, Sighisoara and Bucuresti)

Bucharest to Brasov is between two and three hours depending on which train you catch so entirely do-able as a day trip. Sighisoara is four hours or more. The train to Sighisoara goes through Brasov so you could easily combine the two.

I know Bucharest fairly well but have no first-hand experience of the other two so will leave other people to tell you more.