Italy in March.

There are some sights/experiences that are so in demand that--if you consider them essential--you should get the tickets in advance. These are the ones I remember being mentioned for Rome (not saying they are musts, just that you may miss out on them if you don't buy tickets in advance):

• Borghese Gallery (advance, timed-tickets always required)
• Colosseum Underground Tour (this special-access tour sells out early)
• Vatican Scavi Tour (ditto)
• Vatican Museums are reportedly mobbed these days, and most recent visitors say it's critical to sign up for a tour that gives you early access before the museums open to the general public. Walks of Italy and Dark Rome offer such tours; there may be other options as well.
• Domus Aurea (reportedly open only on weekends and sells out)

There can be long lines for regular entry at the Colosseum, but I don't know what the situation will be like in March.

It was recently reported that the Pantheon will soon start charging admission. That site was extremely crowded when I visited it in March 2015, but I don't know what will happen when there's an admission charge, nor do I know when that policy will be implemented.

Thank you so much, this helps a lot. What online sites would you recommend to purchase tickets?

The official sites are the first choice, but I sometimes use a broker if no tickets are available and I don't have much margin to work with.

Note that some sights provide a guided tour with the ticket price (Domus Aurea, for example).