It's been a while since I was in Paris*, so I'll leave that to others to suggest things. I have been in Rome recently, so here are some things that are good, IMO.

Eating place:
La Piazzetta de Trastevere. (Google Map) This is an excellent place with a wood-fired grill/pizza oven. The gnochi is amazing, the desserts are incredible, and the seafood is wonderful. Trastevere is a part of Rome that's worth seeing, if you get the chance, BTW. If you have time, wander around the streets and see medieval Rome, if not always in the buildings, at least in the street plan.

Things to see:
If you like history/archaeology, San Clemente basilica is worth seeing. The basilica is 12th century, but excavations underneath have uncovered a 4th-century basilica and a 1st-2nd century insula (apartment block).

My wife is rather fond of the Torre Argentina Cat Sanctuary. It's also the site of Julius Caesar's murder, if that's the sort of thing that you enjoy, although the chalk marks drawn by the forum police have long since washed away.

The Forum. But first, read Silver Pigs, or at least watch HBO's Rome. Actually, I found Silver Pigs to be an excellent thing to read before seeing most of Imperial Rome, not just the Forum.

*Okay, I went though the airport (CDG) on the 2nd of January; the only suggestion I have is "Don't eat in the airport."