The day trips: Piza to go see the leaning tower. It's a couple hours by train and then a leisurely stroll from the train station. There is more to Piza than just the tower, a fabulous candy store on the walk over. The gigantic church with it's baptistry next to the tower. The market stalls with the touristy junk for sale right near the tower. The touristy junk is actually priced quite good and I love to pick up interesting items here. Then from the tower we went to the smaller Piza train station and took the train to Lucca. It's an interesting little ancient town. We like to eat at the outdoor cafes in the old roman piazza. Then take the train to Florence to switch to the train to Rome. Definitely a day of walking around and soaking up Italy. Drink a few cappuchinos on the walk to the tower in Piza. There really isn't an reason to be in a hurry. As far as I know the tower isn't going to fall over yet.
Another day trip: Naples area and Pompeii. The first year we took the earliest train to Naples and then hired a driver with a car to take us up to the trail head to hike up Mt Vesuvius. It's a fun hike and at the top you get to look down into the volcano and out over the bay of Naples. It's a beautiful view. On the way down we bought trinkets and wine at the little shop next to the trail. The driver then took us to the entrance to Pompeii. Pompeii is so huge that it really deserves a whole day for it's self, but any time spend there is well worth it. Then after a few hours wandering around Pompeii we had the driver take us back to the train station in Naples.
It's not really necessary to hire a driver if all you want is to go to Pompeii. It's simple enough to catch the commuter train on the lower level of the Naples train station. It stops at all of the local villages and Pompeii also.
I have heard that another day trip that is good out of Rome is to take a bus from Rome to Assissi, but we haven't done this yet.