If you like being overwhelmed by Baroque and art, go to the Palazzo Doria Pamphilj. The gallery within is amazing, the architecture is way over the top!
Also would recommend the National Gallery of Modern & Contemporary Art. Not too busy, because people are elsewhere!
Finally you can't miss the Borghese Gallery. It takes a bit of finding in the park but is excellent. Also the Gardens are amazing.
Of gourse there are ruins. The best (imo) is Trajan's Market, the world's first shopping mall. All the tour groups go to the Colesseum et all, but there's plenty of other ruins to admire.
If you are a bit religious (or even not, because I found it very freaky) take the Metro and see the Basilica di Santa Maria Maggiore. It's full of popes and the odd Bernini item. Pope John Paul the Second was interred there but I think since was sainted he's moved to St Paul's.
Ice cream tip: only visit places with flat ice cream. If its overflowing in the tub, they've whipped it up to make it go further.