Is it best to buy tickets online for places like the Sistine Chapel and the Colosseum online before I leave on my trip or should I buy tickets in Rome, and where?
I think it depends on how felxible you are - and how flexible your schedule is. If you've only got a day here and two days there -- and you want to see a specific site, I think you should lock it in.
Some sites have limited access. The Borghese Gallery, for example, limits the number of visitors (to 200 at a time in this case). DaVincni's The Last Supper in Milan only allows a couple of dozen people in for a few minutes at a time -- and only releases tickets every couple of months. Those tickets go within an hour.
For such sites during a one-, two- or three-week itinerary, buying in advance means you know you'll see them -- and avoid the ticket-buying lines, too.
The Rick Steves guidebook effectively details what sites' tours you should buy in advance, and how to buy them.
The Colosseum has a new ticket system that requires advance reservations, though it may be possible to get a ticket the day of your visit if they haven't all been sold out. Don't take that risk; it often sells out weeks in advance. Buy your tickets online at www.coopculture.it. Also, while you can buy tickets at the entrance to the Vatican Museums, you will have a long, long line to wait in. Smart travelers buy them online in advance of their visit, see www.museivaticani.va.
Wise words above from both T. and gbrennan! Rome's #1 and #2 tourist attractions (Colosseum and Vatican Museums) are seriously overrun these days - ESPECIALLY during high and shoulder seasons - so unless you want to waste a lot of time in long lines or risk not getting in at all, buy your tickets in advance. For both of these attractions you will have to choose a specific day and time slot from what's still available.