Suggestions on where to go and what to do for a few days after Rome

We just did that tour last month and spent 2 extra days in Rome. I highly recommend Ostia Antica. It's a short train ride from Rome. I also recommend the Testaccio area of Rome, especially the Protestant Cemetery (where both Keats and Shelley are buried). Plus, it's the home of the only pyramid in Rome!

You don't say how many days is "a few", or time of year, but a couple of extra days in Rome wouldn't go amiss. Or you could continue southward to Naples or Sorrento to explore those cities and Pompeii, Capri, and the AC.

There's enough in Rome to keep you busy for months, so staying right there is one of several decent options.

I really like Orvieto, which is 1-1/4 to 1-1/2 hours from Rome by train. Current fare is 9.15 euros each way for the slower trains and up to about 15 euros for the faster ones. There are departures from both Roma Termini and Roma Tiburtina. Rick has a good write-up about Orvieto in his guide to Italy. There's a lot to see there; it's not just a pretty hill town.

Castel Gandolfo area is nearby.

Great feedback so far. We will be in Rome in mid-March. By a few I meant 3-5 days.

I second the recommendation for Orvieto. We stayed there for two nights before five in Rome.
We all wished we stayed longer in Orvieto.

You can easily spend all of your extra days in Rome and not run out of things to see and do. If you must, there are several day trips you can do withour having to change hotels.

By a few I meant 3-5 days.

How many nights do you mean?

It really matters on what sort of itinerary is an enjoyable add on and what is just a race to do.

Another vote for a day trip to Orvieto. Such a lovely place and a beautiful cathedral. Underground was also interesting.