The following tours are places I've relished visiting when I'm in Rome. Some you can see by just walking around. But sometimes an organized tour gives you a bit more time to learn about things rather than fumbling with maps and guidebooks.
Tours from Rome to Destinations I like
Ancient Ostia Small Group Day Trip from Rome Ostia was once the ancient port of Rome and the city, founded in the fourth century BC, was abandoned when the port silted up. Great preservation of the buildings. And rather than being a resort, it's a real, working town--so you'll see the apartments of working folks just as they were thousands of years ago.
If you're in Rome and have no intentions of heading south to Campania, you can take an ortganized to Pompeii and Vesuvious from Rome on an air conditioned coach. The tour will save you from having to make train reservations and worrying about schedules. You can get some exercise on the trip by walking up to the summit of Vesuvius where you'll get a very intersting view.
If you want to really feel like you're in Ancient Rome, I suggest a walk along the Appian way and then ducking into the Catacombs when the weather begins to swealter. Nothing like inching along ancient hallways stacked with bones to remind you you're not in Peoria any more. Catacombs and Roman Countryside Half-Day Walking Tour should fit the bill, and it's not expensive at all.
Lots of people like the Rome neighborhood of Trastevere, on the "other" side of the Tiber, so the Trastevere: Trends and Tradition Tour would be right for them.
Customized and Personal Tours of Rome
David Downie, author of Quiet Corners of Rome
Odd Transportation Tours
You know what's popular? The Rome Segway Tour. If you haven't ridden one of these contraptions, don't worry, you'll learn quickly. Even if you're as unbalanced as I am, these things are designed to make you stable, at least for getting around a city at a slow pace. That's why they cost a zillion bucks. You can pretend you're in Roma to make a flick while you're riding. Other tourists, envious--or, maybe they're thinking you're a complete dork--will raise their cameras as one and you can't outrun the clicking of shutters as you glide down the cobbled biways of the Eternal City. Just do it.
For the Scholar in You
If you really want to know a place from folks who've lived it and studied it, for my money Context Tours are the best. I've taken tours from then on many occasions, from Rome to Istanbul, and they've all been excellent.
For Foodies, there's Savoring Rome, A Culinary Stroll
Reall, there's too many fine Context Travel Tours to list, so here's the page on Rome: Rome Walks including the Underground Rome walk.