There is a moment on any of these flights where the landscape suddenly explains itself. The lakes stop being lakes and become the holes left by volcanoes. The Appian Way becomes a straight line across everything. The mountains behind Tivoli turn out to be a wall, and the plain in front of it turns out to be flat because the sea used to be there.
The options divide by how much you want to be involved. A balloon is silent and does not require anything of you. A tandem paraglider needs you to run a few steps off the edge at Norma, where the Lepini escarpment falls straight to the plain. A skydive requires the sort of decision you can only make once.
All of it depends on weather, more than anything else on this site. Flights get moved. Book early in your stay so there is room to move them to.
What you actually see from up there depends on which side of the city you go up. South, the two crater lakes sit inside the collapsed volcano with the Castelli towns around the rim, and Rome runs out to the sea behind them. North, the Tiber valley and the vineyards, with the balloon flights going up at first light, which is when the air is calmest. West, the coast and the fish farms. From over the Lepini, the whole reclaimed plain is laid out to the sea like a map of itself.
Everything here is priced and timed by the operator and almost all of it is weather-bound, which changes how you should book. Put it early in your stay rather than on the last day: a flight that gets moved is normal and a flight that cannot be moved is a lost booking. The operators on this page will tell you the evening before, and the ones who fly regardless of the forecast are not the ones you want.
Weight limits and minimum ages are the other constraint and they come from the operator and their insurer rather than from us. They are on the card where they are published, and where they are not, a phone call before you drive out is cheaper than the drive.