Smoking is banned in Italy in all public places. This includes trains, buses, offices, hospitals, and waiting rooms, as well as restaurants, pubs, and discotheques (unless the latter have separate smoking rooms). Fines for breaking the law are exorbitant. You'll find that most people skirt the law by sitting outside on the many outdoor restaurant terraces. Sit inside if the smoke in outdoor seating areas bothers you. Many restaurants are now equipped with air-conditioning.
It's always best to travel with your own trusted medications. Should you need medication while in Italy, you should speak with a physician to make sure it is the proper kind and in case a prescription is necessary. Aspirin (l'aspirina) can be purchased at any pharmacy, as can over-the-counter medicines such as ibuprofen or paracetamol. Other over-the-counter remedies, including cough syrup, antiseptic creams, and headache pills, are sold only in pharmacies. Pharmacists are happy to dispense advice and, in the city center especially, almost always speak some English.