The two main things we did in Rome on a Sunday this past June were:
Domus Romani at the Palazzo Valentini (#1 attraction on Tripadvisor and wonderful!)
San Clementi Church near the Colosseum (but be careful, it closes around lunch time), also very very interesting.
The pizza shop across the street from the entrance to San Clemente was our favorite pizza of our trip.
We also had gelato, of course!

I'm about to write about our Sunday in Rome in a trip report if you want more detail:

http://www.fodors.com/community/europe/top-chef-abruzzo-a-week-in-an-italian-cooking-school-rome.cfm

It might take a couple of days to get to that part, so hang in there with me!

The only place that's closed on Sunday, of interest to tourists, is the Vatican Museums (except the last Sunday in a month when it has free entry and is madly crowded). Also not a good day for shopping, as most shops are closed except some in the centre.
All ancient sites are open - Colosseum, Palatine and Forum (same ticket). Museums too, though often AM only. As are churches (no sightseeing during Mass). Gelateria are also open, and long hours (often to midnight and beyond in the popular areas).

ibatkins, I'm sure that you'll find enough to do to fill a day in Rome on a sunday. Latin countries have a different attitude to sundays than protestant ones, and in fact the worst day is usually Monday when often a lot more is shut than on Sunday. We were amazed that in Venice on Easter sunday, virtually nothing was closed, including the Accademia.

so by all means check on opening times, but don't worry too much about not finding things to do.

and if you want gelato, after your visit to the Colosseum etc, walk up the via dei Serpenti - just ahead of you as you head north past the forum - and you'll find and excellent gelateria on the left, and before that, on the right, a very nice square with a fountain, bar, cafe, and lots of nice local restaurants offering sunday lunch. Keep on walking and you will get to the via nazionale and can catch a bus [45 or 60] down to the Victoria Emanuale monument or stay on longer to get to the Campo dei fiori or the Pantheon.

have a great trip!

Thanks for the replies! I'm relieved there will be plenty open, some of the sites I was reading had me fearing the worst!

the galleria borghese gallery is open on sunday and, i think, closed on monday and it is just fantastic. also-- the park that surrounds it is a great place to spend time on a sunday afternoon, as there will be lots of locals out on a stroll.

i know you want gelato and more power to you-- but if you like coffee, i recommend that you also have a granita at cremeria monteforte next to the pantheon. "granita di caffe con panna" is sooooo delicious... the strongest most delicious coffee granita layered with thick fresh whipped cream. mmmmmmmmm.