The obvious day trip from Najaf is Karbala, eighty kilometres north, site of the shrines of Imam Husayn and his half-brother Abbas — the second great pilgrimage city of Shia Islam. Shared taxis run constantly. Many visitors do Najaf and Karbala back to back as part of a single pilgrimage.
Further north, Babylon can be visited as a long day trip if you set out early. The ruins are a shadow of what they once were — much was looted, much was rebuilt in modern brick under Saddam — but standing on the foundations of the Tower of Babel is still something. Closer to home, the ancient city of Kufa, with its great mosque, is so close it counts as a half-day rather than a full day trip.