Qom is one of the holiest cities in Iran and the middle East and is entrenched in centuries of history. The famous thing about Qom is the Fatima È Massummeh Shrine which is a highly respected shrine and a very clean place. Non-Muslims are allowed entry to the city , but they are not allowed entry in the holy shrine unless with a Muslim companion or guide.
Qom is the main city for religous studies in Iran for long. Hozeye-Elmiye-Qom is the largest Theology school in Iran. Right now many Senior ranking clerics of Shia Islam live in Qom.
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: 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 shops. Qom is famous for it's "sohan," a flat sweet biscuit made of pistachios and saffron. "Gaz", a nougat sweet 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 compact discs.
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. Most of the restaurants in good quality of food and environmental conditions are in Mohamad Amin Blv. which is recognisable by its tall compact trees. Ranking from top, you can find 'Yazdanpanah historical house restaurant  Arg Salariyeh, Alborz, Morvarid, Jam, Markazi. You can find Kabab, grilled chicken and fried fish there. Dizi baar (to eat Abgoosht)and lebanese (to eat labanese food)restaurants are in Ghadir Blv. that is along the Amin Blv. There are also a lot of fast food places such as Bama, Shahre Ma, Fadak, and Dolfin where you can eat different varieties of sandwiches and pizzas.
In Qom, there are various cafes and fruit juice shops. Pomegranate juice is popular and is said to heal the sick. In the winter when it snows, hot chocolate and Persian tea (Chai) are available. Dast'gheyb square is a known place where you can find any non-alcoholic drink(fruit juice, faloodeh, coffee, hot chocolate, milkshake, etc.) and ice cream. The best place for having Faloodeh is a place called Faloodeh Momtaz located near Motahari square (exactly located at the intersection of Mowlavi street and Hadaf street).
Qom is one of the holiest cities in Iran and the Middle East, and as such be aware that Islamic customs and values are more likely to be important here than elsewhere in Iran. Don't let this worry you at all, just ensure that you are respectful of religious and cultural practices and that you are dressed appropriately and modestly.
If you feel like visiting another quintessential Persian city, a few hundred kilometres away, you can visit Yazd which is an beautiful ancient city. Isfahan is another option and is a very beautiful city and is considered to be one of the oldest and most historic cities in the world.