Is it cheaper to book train tickets on line before my trip to Italy this summer, compared to buying tickets while I'm there? If so, which website?
I tend never to buy train tickets ahead of time since I think they are more expensive if buying in the US on a site like raileurope. The exception would be if you are going for a railpass which isn't sold in Europe.
In general, I tend to go for the point to point options as I don't think the railpasses are economical if traveling outside of France, Germany, Spain.
Unless you're going to take really long trips, buying once there is easier and more flexible,
Does that still apply when traveling in July? Isn't a ticket in hand peace of mind during the busy season?
We will be going between Venice-Florence-Vernazza-Nice.
It all depends on when you are going and which trains. You can get advanced discount fares such as the AMICA fare by booking even in Italy. But they have to be purchased at least the day before train departure and the number of discount fares per train may be limited so it may be advantageous to buy on line. If you are going to buy on line use the Italian railway's website: http://www.trenitalia.com/en/index.html.
Note that when you are able to take advantage of discounted tickets online (Amica, Smart, etc.) early, they come with restrictions. Usually, no refund and no changes. Thus, you may be locking yourself into something you can't change. As mentioned above, these discounted tickets may (or may not) be available while you are there if you buy 2-3 days early. Amica fares are limited based on day, train and class of train.