From the airport
You essentially have three ways of getting from the airport to your destination. What might surprise you is that Taxi is usually the cheapest solution but it depends on how many you are, where you are going etc.
Readily available outside the terminal at Arlanda they drive on fixed prices to any destination in central Stockholm. The prices vary a bit depending on the company but should all be around 470-500SEK and the price should be written on the outside of the car. Don't ever go by meter, the prices are fixed! If you are more then 4 persons you might want to consider preordering a large taxi that takes up to 8 people, ask your host to help you with this.
The fastest and most environmental friendly way of getting to central Stockholm. It takes about 20 min to get to central station and the trains run all the time. Unfortunately it's expensive so usually it doesn't make sense if you are more then 2 persons but there are exceptions. Usually they have a weekend special where you can go 2 adults for less and a summer special where children go for free etc. Problem is that once you hit central station you will still probably want to take a taxi to your destination.. However, if you are going to Arlanda during rush hour, this is unbeatable! To see the current prices please check out their webpage where you can also buy tickets in advance: Arlanda express
The cheapest way of getting to central station. It takes quite longer then a taxi due to all the stops and they don't run as often as the trains but is a cheap way of traveling. Again, you will however be dropped of at central station and have to make your way from there to your destination. For prices and schedule, check out their webpage: Airport busses
First things first, Skavsta is quite a long way from Stockholm despite what Ryanair says so don't be surprised if the journey takes a while. From here I would only really recommend travelling by bus. A taxi is ridiculously expensive so this is a big no no if I were you.
The busses matches all arrivals/departures so you rarely have to wait for it. A good idea is to buy a 2 way ticket right away since it's cheaper and it saves you the hassle of doing it twice. For schedules, prices and more, check out there webpage: Airport busses
Public transportation in Stockholm is really good if you don't happen to be here during the winter when the occasional snowstorm brings everything to a stop.. It's run by SL and if you are here for a few days you can get a visitors pass that are good value. When you get a pass you can go on the commuter trains, the metro, all the busses and trams plus a few boats so it's a really good idea. SL has got a good visitors page that you can reach from here: Public transportation in Stockholm
Stockholm City bikes or as they are more commonly known: "Alvedoncykel"
The advil bikes.. Yes, sponsored by the Swedish version of Advil, theese bikes are readily availble all over Stockholm. They are cheap, really smart way of getting around and good for the environment. Pick up a bike at one stand, drop it of at another, never have to worry about locking up your bike or where to store it at night. Join the movement! See their webpage for prices and more info: Stockholm City bikes