What a great opportunity, sounds like it will be an awesome adventure.
The train pass will probably suit your needs quite well - I'm a bit out of date with the latest charges and what not, but generally speaking, you only need to pay extra reservation fees and such on the express trains, for overnight berths, or to guarantee a seat on a busy route. Generally speaking you should be able to just hop on a slower train with no extra fee, though you can't be absolutely guaranteed a seat in those cases.
Generally the passes work out to a good deal if you use them on longer distance routes, so you may run into extra costs for transport within a city or it's surroundings. However, you can often buy local transit tickets covering anywhere from a day to a week which can be quite a good deal, as they usually cover areas outside the metro centre where you might want to take a day trip or two.
It's also a great time of year to be travelling, as the weather should be gradually improving and should be just beautiful in Italy in April, and still not too terribly crowded. The main snag that time of year is the Easter holiday - usually the week before and the week after Easter itself, when every European school child is on holiday and many many people cope by taking a vacation. This can lead to crowded hostels and budget hotels (hopefully not a big issue for you), and sometimes the trains and cheap flights will be stuffed to capacity as well. Not to mention tourist hordes at the big attractions. Unless you have always wanted to be at the Vatican for Easter, it might be wise to opt for a smaller community during that time - you'll still get to see lots of local customs and enjoy traditional events and foods, but without quite the throngs you'd get in Rome.