The city of Stockholm is located on fourteen islands and the most interesting bits lie on four or five of the islands. Your options for getting around include the Tbana (Tunnelbana or subway), the busses, the Pendeltåg (J commuter train), automobile, bicycle and walking. This ignores crawling around drunk as the general view of such a mode of motation here is that it's inefficient and generally obnoxious. Same goes for skating, skateboarding and dogsledding in the city limits. I will try to cover the acceptable means of transport in more detail.
Public transport in Stockholm generally gives you most of the mobility you will need for most purposes. If you plan to stay for any length of time, you will probably wish to buy general tickets by the month or greater duration of time. A one day pass good on all Tbana lines, busses and commuter trains currently costs 60kroner. A three day pass is 120kroner. A thirty day pass is 400 kroner. A four month pass is 1550kroner and so on. As you can see, the big price break comes with the thirty day pass.
Using your SL kort (public transportation ticket), you can take commuter trains, Tbana and busses, or any combination thereof to get to most destinations in and around Stockholm. The Pendeltåg, or commuter train, has two lines that go north and south generally. The Tbana has three lines that go generally north and south. The busses go everywhere. In later parts of this set of web pages, I will refer to combinations of Tbana routes, bus routes and walking to describe how to get to various museums, restuarants and other desirable destinations.
The central bits of Stockholm are fairly compact and so, during good weather, walking presents a great opportunity to learn the lay of the land and to exercise your lungs and legs at the same time. During my stay, I lived in Södermalm, or south island. It's not an extremely long walk from there to the city center via Gamla Stan. It's also like a walk through history. To do this, at least the first few times, a decent map makes sense. I suggest Stockholm's Official Tourist Map for 15SEK. It's not too big and shows decent detail. Sometimes walking is the best available option, as when the Tbana has closed but you have been out with friends. If you are within easy walking distance, a stroll home might clear the alcohol from your head.....finding a cab may be a bit of a challenge in such cases.
Taking a cab is often a good option when you are either too drunk to drive, too drunk to walk or when the weather truly sucks or when you are going somewhere where no bus-route or Tbana stop is convenient. Taxis also have the advantage of all-night availability. I recommend either Taxi Kurir(08-300000) or Taxi Stockholm(08-150000). If you really need to get to the airport (Arlanda) in a hurry from your front door, taxis are the way to go. Also, if you've been partying with a group of friends and you just don't have the ability or patience to walk, or take public transport, then taxis will gladly pick you up and deliver you to your destination--for a fee, credit cards accepted of course.
Stockholm is a good city for cycling, except for the weather. Really, I cannot think of a better way to explore. You can't go too fast, but you can cover a lot of ground. There are bike paths all over the city but pedestrians often ignore them and just stand around like complete idiots in the middle of bikepaths. (In all fairness, this also happens in other places as well. In Austin, for instance, bikepaths on the streets seem to attract people as parking places...). In any case, if you plan to cycle a lot and stop anywhere, get a good lock. In my explorations of Södermalm and other bits of Stockholm, I saw more neat shops, restaurants, parks and interesting sights from my bicycle than any other way. I heartily recommend cycling in Stockholm.