Stockholm has three main airports: Arlanda, Bromma and Skavsta. There is a number of ways to get into the city from them. These include Train, Bus, Pendeltåg (commuter train), Taxi (or Uber).
From Arlanda Airport
At Arlanda, once you've exited the luggage pick-up point, there are signs directing you to either taxi or train/bus.
The train is operated by the airport, called Arlanda Express, which, depending on your age, costs between 150 and 280 Swedish Kronor (SEK) per person. If you buy the ticket online you can get a discount up to 45% off. See more details at https://www.arlandaexpress.com/priser.aspx. The trip takes 20 minutes, the train departs every between 3 and 6 times an hour from both ends and makes a stop at Terminal 5 (international) and Terminal 4 (national) as well as Stockholm City (T-centralen). After 10.00 in evening the times might be less frequent, but at least twice an hour until 01.00 in the morning. See their full times here.
Taking the bus there are several buses going to different places in Stockholm Region, as well as to Uppsala, a city north of Stockholm, and if your main goal is to get to Stockholm City, there are buses operated by Swebus and Flygbussarna, costing between 89 and 119 SEK, depending on age and company. The ride takes about 40 minutes and will land you at the bus terminal of T-Centralen in Stockholm City. Taking taxi is the most expensive alternative, but it may be comfortable if you have a lot of luggage. It can bring you wherever you want to go, if you have a hotel etc. but if you are going to T-centralen , the price will be about 500 SEK.
Commuter Train and Communal Bus
The cheapest alternative to get into the city will be by using SL (Stockholms Lokaltrafik) by catching either the Pendeltåg, which is a commuter train operated by SL (Stockholm Lokaltrafik), which will take you to the train central at T-centralen, or the communal bus number 593 which gets you to Vasagatan, a centrally located street in Stockholm City. The ticket costs 30 SEK and is valid for 1h15 min. Both rides takes around 43 minutes. However, taking the Pendeltåg you will be forced to pay the 'leaving fee' of leaving the airport, which is about 79 SEK, so the bus might be a better alternative.
From Bromma Airport
For arriving at Bromma, your options are Taxi (or Uber), bus (Flygbussarna) as well as communal traffic (Bus by SL), which will get you to where you need to go, typically T-centralen, for between 30 SEK (SL) and 300 SEK (Taxi). Flygbussarna costs 75 SEK.
The Taxi services as well as the busses stand outside the airport, right after you exit. The buses don't come too often, as it is a much smaller airport, but a few times an hour. They take around 20 minutes to get to the city, and are quite comfortable. The Taxi will take around 15 minutes, so it isn't that much of a time-save if you're not looking for getting there with a lot of luggage.
From Skavsta Airport
If you're arriving at Skavsta, this is located quite far away from Stockholm City, and buses leave typically only when flights arrive, which means after you land you will have approximately an hour (but probably less) to get to the bus before it leaves, and when it does, you will have to wait for the next one, which probably won't be around for another couple of hours. Taxi from here will be expensive (according to most taxicompanies, around 1000 SEK), and take around 1 hour, while the bus is about 139 SEK and takes 1hour 20min, so we would recommend the bus.
For more information, see the websites of all the operators:
Public Transportation in Stockholm
Once you have arrived at Stockholm City (T-centalen) you are free to go wherever you need to go, but the metro stops running at 02.00 in the morning every Sunday to Thursday, while on weekends it runs all night. Buses run all night, but less frequently during the night.
Getting around in Stockholm is easy, there are a number of ways, bus, metro, commuter train (pendeltåg), bicycle, tram and boat. Most of these are operated by Stockholms Lokaltrafik (SL), which means you can buy a ticket for 30 SEK lasting for 1hour15min, or an SL Access card (which costs 20 SEK) and charge it with anything between 24 hours and a year which costs between 120 SEK and 8720 SEK depending on age, validity and whether you're a student or not. The most popular for students is getting a monthly card for 550 SEK.
SL Access Card
The SL cards are sold by various shops in Stockholm, including the SL offices, located in most subway stations, and other shops such as Pressbyrån, 7-eleven etc. They can then be charged either in the shop or in the machines in the metro and buss stations.
Metro and Bus
The transport lines with SL are several, for the metro there is the green, red and blue lines, which all split up in different directions, adding up to the total of seven metro lines. They run throughout the main parts of Stockholm region, and the rest is covered by buses. Buses have different numbers based on where they are going, and taking the right one might seem confusing at first, but you learn quite quickly. Tram isn't very common nowadays, but it is a nice way of experiencing the center of Stockholm while inside above ground. See more information about tickets and prices at SL.se/en/fares.
The commuter train travels the furthest distances and is a great way to save time rather than going these distances with bus. There are 4 lines consisting of a total of 53 stations, and on average it takes about 3 minutes between each station.
Bicycles and boat
Bicycles are around to rent at different stations in the city, operated by Citybikes. You can buy a three days card for 165 SEK or a season for 250 SEK. You can also rent it with your SL card.
Some excursions to different islands are done by boat, which will have different prices depending on where you are going, but at some of them you can use your SL card.
Going Outside Stockholm
If you want to go outside Stockholm to travel to another city or country, you will have to use SJ (or plane/boat). The trains are good and quite reliable, and offer good services as well as a nice scenic view over the passing landscapes. Prices can be found at SJ.