Flights- Frrom India to Rome are often cheap, specially to Rome and Milan.
March, April, May, September, October are good times to visit if you want to avoid heat and sharp rise in airfare of peak summer months in Italy. In March/April we found Bangalore- Rome return flights for around Rs.40000 per person
Stay -There are many options in Rome to stay in terms of hotels, resorts, Airbnb, hostels etc. It is better to stay close to metro stations .
Transportation - Metro fare is cheap around 1.50 euros per ride within the city limits and locals use it all the time. Buses are also plenty and a little bit cheaper than metro. If using metro, you can but a carnet of 10 tickets at once to avoid buying every time.
Beware of touts and pick pocketing- just like you would be in most major cities around the world , specially in trains, buses, crowded places, ticket queues.
Sights - If you are visiting the Vatican, Pantheon etc, consider buying tickets online before hand as there are huge lines in summer months or other peak times. In shoulder season like March/April/October/November, expect a wait time of 15-50 mins in the line of you buy on the spot.
Eating- Ask about coperto or table charge restaurants would normally apply on bill per person basis. Tips are not expected in this case but you can round up the bill and leave something for the wait staff if the service was truly exceptional. We did it once in Rome.
For more on our travel through Italy read here -
Wine region of Assisi, Italy-
Verenna on lake Como, the lake district of Italy-