To start your journey, buy a travel card at a kiosk (most likely called Pressbyrån) near the ticket barriers in your nearest T-bana station. You cannot buy one from the ticket gate officer or at a ticket vending machine. Don't worry. A shop attendant will understand what a "travel card" is. The price as of June 2008 is 690 krona for 30 days, 260 for 7 days, 200 for 3 days, 100 for 1 day. A travel card allows you to take any T-bana lines and buses in Stockholm for an unlimited number of times while the card is valid.
Alternatively, you can buy a single ticket (the price for a single journey within Zone A is 40 krona if you buy one from the ticket gate officer and 30 krona from a kiosk, as of July 2009) or a pre-paid strip of coupon tickets that allow you 8 single journeys within Zone A (the price is 180 krona, as of July 2009, both at the ticket gate or at a kiosk). Beware, however, that the ticket gate officer is often absent.
When you exit from a T-bana station, you don’t need to show your ticket. Just push the bar of a ticket barrier to exit.
Life with a travel card is much simpler. First, a travel card allows you to swipe the ticket to pass the ticket barriers. Second, you don't need to worry about anything when the station staff at the ticket gate is absent.
If you go for single tickets or a strip of coupon tickets, you need to understand the following rather complicated system of Stockholm public transportation:
A single journey requires two tickets within the same zone, three for two zones, and four for three zones. (See this route map for zone boundaries. As long as you take T-bana, it's always within Zone A.) One ticket costs 20 krona at the ticket gate or 15 krona at a kiosk. The prepaid strip of coupon tickets gives you 16 tickets.
When you pass the ticket gate, you need to have your tickets time-stamped by the staff at the gate. Since there is only one person at the gate in most stations, there is often a queue for having a ticket stamped. What's confusing is that this stamp guy or lady is often absent. In such a case, you can simply pass the gate which is unlocked. But you need to remember what time you enter the station. In a very rare occasion, you'll encounter ticket inspectors at a station exit. If you don't have a valid travel card or a stamped ticket, you'll be fined with a hefty amount of money. By reporting the time you enter the station, you can avoid the fine even with an unstamped ticket (according to my Swedish colleague).
Once you have your ticket stamped, it's valid during the following 60 minutes. So, for example, you can take a bus immediately after getting off from a T-bana station without having more tickets stamped.
Understand? Life with a travel card is much simpler, isn't it? For more details on tickets and fares, visit SL's website in English. (SL is in charge of Stockholm public transportation.)One good (or bad?) thing about T-bana is that you can make and receive a call with your mobile phone.
Many T-bana stations feature art installations, particularly Tekniska Högsoklan and Kungsträdgården. At Solna Centrum, the whole decor of the station is art. Here is the official guide for art in T-bana.
You cannot buy tickets from the bus driver (another reason for why a travel card makes your life simpler). Bus lines 1, 2, 3, and 4 run frequently (every 5 minutes) and can be very useful depending on where you live. For bus route maps, see this post.