Fen,
I'd recommend using several locations for a 2.5 month stay, as that will give you an opportunity to explore different parts of Italy (each will have their own "character"). You stated that "we don't have a huge budget" so using one location and then taking day or overnight trips is not a good idea, as that will increase your transportation costs.
I normally travel solo, so I don't rent apartments. There have been many posts here from people that recommend VRBO but I can't provide any first-hand information on that. I'm sure the others will have lots of good suggestions.
If this is your first trip, I'd suggest starting in the north and working your way south to Rome. You could (for example) fly inbound to Milan, Venice or Florence and then stay in or near those locations. Move south to somewhere near Siena, Cortona or Orvieto and then finish in Rome. While in Rome you can take a day trip to Pompeii if desired, although it will be a full day.
Travel by train (or Bus in some cases) is a good plan, as that's often the cheapest and most efficient method. However, there are some potentially expensive "caveats" with using both public transit and rental cars, so be sure to do your homework on that.
You might find it very helpful to pick up a copy of the Italy 2014 Guidebook, as that has an excellent description of many of towns you may want to stay in, as well as the sightseeing attractions. Do you have any ideas on place you're really interested in seeing?
Good luck with your planning!