Why do you need a tobacco shop if you don't smoke? A Tabacchi is where you go for local bus tickets. Many carry phone cards (scheda telefonica), which are generally the cheapest way to phone outside the country of Italy. You might also find postage stamps (francobolli) at a Tabacchi. A large Tabacchi might sell pens, lighters, watches, candy and jewelry as well.
And yes, you can get cigarettes, cigars, and other tobacco products at a Tabacchi in Italy.
Tabacchi in Italy display the sign you see to the right, with a big white "T" on a dark blue or black background. Notice that the sign says "sali e tabacchi" which refers to two products that were controlled by the government, salt(s) and tobacco(s). While salt was once a government monopoly, it has recently been removed from government price controls. The signs have not changed, however.