Definitely try to get to the Pantheon, my favourite building in the World.

Try putting your camera on the plinth of the statues (no tripods) for really long exposures. If the people keep moving, a lot of them will disappear from the exposure.

Try and get there around mid day'ish' to get the light falling close to the centre of the floor through the Oculus (opening) in the ceiling. If you have a wide enough lens (I didn't) then you can get the opening and the people on the same image to emphasize the height of the ceiling.
You can get some nice night pics of the Colosseum from across the road above the Metro, and from a raised section beside the Forum.
There are some nice views of the Vatican from the Ponte Umberto I (bridge) over the river.
Also good views of the Castel Sant'Angelo from the bridge in front of it, and from the bank opposite. Also good views of the Vatican from that bridge too.
There are great views from the top of the Cupola (dome) of St Peter's Basilica and the square below. It is a trek getting up there, and if I go again, I will video my accent as I didn't think it was as going to be long as it was.

I videoed coming down, but it's not the same.

The angle of the walls got quite 'crazy house' at times.
Here's some pics I took on my holiday there last July
PS If you find yourself near the Mausoleum of Augustus, there are some little arty/quirky/jokey things which have been set up along a wall next to it which are quite interesting. But there is one very interesting thing which made me dip into my pocket and leave some money for the artist. I don't want to say too much as to not spoil 'it'.

I can't guarantee the stuff will be there though.
