Hello, we went fairly recently.
My toppest tip is to book either a guided tour and / or skip the line tickets to the places you want to go. The queues for The Colleseum are AWFUL! But it's an amazing place to see, we all loved it (it was my mum, 72, a 17 year old, 11 year old, me and dh).
We went to The Colloseum with Rome Walks and paid a few hundred quid IIRC but for that we went straight in and had a 3 hour 121 tour with an art historian. It was fabulous, the highlight of our stay.
The Sistine Chapel, again buy skip the line tickets (although you still have to mess about inside getting another entry document) as the queues are very very long indeed. You get shuffled along lots of corridors with lots of other people though, until you get to the actual Sistine Chapel and then it's PACKED and manned by guards whose job is to SHUSH you really loudly! Sometimes using a megaphone, I'm not joking 
The Spanish Steps are lovely if you ignore the rose sellers. We were there at dusk too and had a drink at an outside bar at the top, lovely.
There's no point in going to the Trevi Fountain as it is closed and you can't see much. The Pantheon is nice and starts to light up at dusk.
It's a beautiful city to just walk around. IIWY I'd stay as centrally as possible, we stayed too far out and I wouldn't do that again. I'd look at Hotel Russie and Hassler I think if I went again.