Oviesse,Coin are two department stores that come to mind.

Since you will have a car, it may be easier to wait until you leave Rome and check out the bigger shopping centers in the outskirts.
One chain store which you can check for locations would be www.mediaworld.it

They are called "microonde" and if you are thinking of using them in your hotel room, you shouldn't do that until you have confirmed with the hotel that you won't blow the circuits or otherwise injure the wiring. This is especially true if you don't plan to stay in modern hotels. Be aware too that Italian plugs very often need adapters in order to fit into outlets. There is no standardization throughout Europe of plugs/outlets.

If you are renting a campervan, most have a microwave in them.

You can look online to compare prices and brands:

http://www.kelkoo.it/c-146201-microonde.html

Depending on what route you take out of Rome, you can google the word "microonde" plus the name of the town where you are going (i.e. "microonde Montepulciano")

http://www.promoqui.it/montepulciano/negozi/wellcome/#

Giving how heavy and expensive microwaves are, and how often you are likely to encounter hotel owners who tell you that you cannot use them in your room, you might do better to look for hotels with kitchenettes that have microwaves, which are fairly common throughout Italy.

You are going to find good prices at a Carrefour.

Think of a Walmart / Target combined. The major stores are huge.

I've used them all over Europe, as well as Turkey and Singapore.

Great also for wines and snacks and ....