Qom is a province in Central Iran.
The nearest and most easily accessible international airport is Tehran's Imam Khomeini International Airport (IATA: IKA, ICAO: OIIE), which is due to replace the older Mehrabad International Airport (IATA: THR, ICAO: OIII).
Driving from Tehran might be a reasonable option, however, if you're worried about driving in Iran you can reserve a private bus to take you to Qom. After all, it is much safer to travel by private buses.
There is a pretty impressive bus and taxi system, which has many bus routes to Tehran and other cities. It is very cheap to travel by local buses in Qom. Private tour buses are generally much more expensive.
Qom has many carpet and sweet (candy) shops. Qom is famous for it's "sohan," a flat sweet biscuit made of pistachios and saffron. "Gaz", a nougat candy is also available. Qom is also known for the unique and beautiful silk rugs which are famous worldwide for their unique silk and patterns. Naturally, as one of Islam's holy cities there are numerous religious shops which sell religious books, versions of the Koran and cassettes.
There are lots of places to eat in Qom which range from old style Persian restaurants to international restaurants. Persian restaurants offer a more authentic cultural atmosphere than international restaurants.