The colosseum, or the Flavian Amphitheatre if you want to be pretentious, is perhaps the most iconic Roman ruin in the entire empire. To avoid the queues we arrived early, taking full advantage of our student discount for the triple ticket which included the Roman Forum and Palatine Hill. After a lacklustre tour from one of the official guides we explored on our own, guidebook and personal knowledge in hand. Whilst many of the walls remain the seats and steps are gone, meaning imagination must be used. For me however the highlight of this day was the Roman Forum.
As someone who studies ancient history and spent two weeks researching and writing an essay on the Forum Romanum, this was an exciting day. The forum itself is large and has ruins from the very beginnings of Rome through to the end of the empire and beyond. Unfortunately the majority of the buildings in the forum are now ruins but you can go into the ones that were turned into churches, and thus saved from ruin. The signs aren't too detailed so I would recommend getting a good guidebook or a tour. I could have spent all day in the forum, just imagining what it was like in the glory days of Rome.
The Palatine Hill overlooks the Roman Forum and is the site of some ruins as well. The beautiful Farnese gardens provided a shady spot on a sunny day. It is the views of both the forum and the colosseum that make the uphill walk worth it in my opinion.
The Capitoline Museums have a good collection of art and archaeology, making it a good introduction to Rome. There is a large amount to see but the layout and variety means you can spend a good couple of hours there. For me the best part was underground in the Tabularium, in archaic times this is thought to have been archives and nowadays has an impressive view of the Roman Forum.