I think you want to say "The library begins at ten o'clock" and "The museum begins at nine o'clock." In English the correct way to say this would be: The library opens at ten o'clock. The museum opens at nine o'clock.

Library beings in ten o'clock. (The library opens at ten o' clock.) Museum beings in nine o'clock. (The museum opens at nine o' cock.) I stayed the hotel with my family. (I stayed at the hotel with my family.) Ms.Miki teaches science. (Ms. Miki teaches science.) She called me Mai. (Good) They helped each other. (Good) He caught the taxi. (He got/took a taxi.) I caught a butterfly. (Good) I swam in the sea. (Good) He swam in the pool. (Good)

I think you want a different verb or conjugation than "beings". Also, you need the preposition "at" in the phrase: "I stayed **at** the hotel with my family. good luck!