Outside of Rome
Heading south, the Castelli Romani wine country (Nemi, Frascati, Monte Compatri), all about 20km away. Trains from Roma Ostiense, I believe. Fatrher afield, Sperlonga and Terracina on the coast.
North. Lago di Bracciano, itws most picturesqe town Anguillara Sabazia, also by train from Roma Ostiense. Farther afield north, Viterbo and its medieval quarters..
All of these places have good tourist web sites.
Take the train to Orvieto (just over an hour). Spend the morning visiting the cathedral and strolling the very picturesque town and find a wonderful place to have a nice Umbrian meal. If you like the sound of this, search this board for Orvieto and you'll get lot's of suggestions about where to eat.