Get nearly everywhere by train. Book a train ticket: SJ
SJ also offers student discount
Check out Ybuss. Covers the entire nation and is also perfect for a ride to the Arlanda Airport
You know exactly how this works... ;-) But in case you don't, here are some hints: SAS, Norwegian, RyanAir and the Star Alliance.
You can fly to Stockholm, Gothenburg and Luleå from Sundsvall-Timrå Airport. SAS flies to Stockholm-Arlanda (Sweden's main airport), Bra flies to Stockholm-Bromma and NextJet (not existing anymore) flies to Gothenburg and Luleå.
Tired of seeing cities and tourist spots? Rent a car and explore the countryside.
Go to: Renting a car
You cannot pay with cash for any public transportation!
Sundsvall and other smaller cities allow you to pay with credit card.
BUT bigger cities (with a proper metro and bus system) require you to get a metro card. When you arrive simply go to the nearest Pressbyrån and ask them for a metro card. There you can also directly charge the card or add e.g. a 1-day, 3-days, etc ticket!