We are flying from Atlanta to Rome, and will be going from there to Pisa (relatives live there). Better to (a) fly, or (b) take a train?
Depends. How much extra is the plane ticket extending to Pisa. What is the air schedule? Train will take 3 1/2 to 4 1/2 hours. Two or three connections every hour except for the dead hour between noon and 1 pm. Cost in the 30 to 70 euro range depending on the type of train and time of day.
Cheapest option on Delta looks like ATL>JFK>PSA, bypassing Rome. Only available after June 1.
Definitely take the train. If you book early enough you can get tickets for €9 per person, The train ride was very pleasant and my kids had so much fun. Plus you're going from city center to city center.
It was not clear from your post whether Pisa is your final destination on the same day. If in Rome for a few days, definitely take the train.
Keep in mind that flying will involve waiting for your flight and perhaps another security check before boarding. You can compare travel times at rome2rio.com.