1

Welcome to Sweden. Taxis here are expensive so I have never caught one to or from an airport here .. but I tend to travel solo so it may be more economical if there are 3 or 4 of you travelling together. The Arlanda airport page gives maximum taxi price of 625SEK but I tend to go by bus or train into town as it is cheaper, train is fastest and bus is cheapest.

http://www.swedavia.com/arlanda/to-from/taxi/
http://www.flygbussarna.se/
https://www.arlandaexpress.com/default.aspx?page=3

For your travel between Stockholm and Gothenburg I tend to go by train, quicker on the express at just over 3 hours than about 7 by the bus. With the trains the tickets are released 90 days before travel and are cheapest then, booking the day before is more expensive. There are many trains per day between the 2 cities and the type of train and length of the journey can vary by a couple of hours, get the XJ trains, express and quick :-)

http://www.sj.se/

2

Taxis, as stated, are expensive. From Arlanda to the city count SEK 550 as minimum. For comparison, for bus ticket one way you'll pay SEK 99.The trip takes 40 min.
For hotels you'd need to state your budget. Prices in the range SEK 500 to 2000 for a single room. Hostels from SEK 220-250.

3

Basically, there are two train routes and one bus route (about three bus companies).
-Faster trains (X2000), about three hours, SJ
-Intercity trains, 4-5 hours. They travel a slightly longer route via Vasteras, Orebro
-Buses/coaches, usually via Norrkoping, Linkoping, Jonkoping, 6-7 hours