Public transportation ticket
You'll need an app named "Västtrafik To Go" to buy your ticket.
You can buy tickets on the second page.
There is one time ticket, one day ticket and one month ticket.
For the one month ticket, if you have done with verifing your student status, you can buy student ticket.
Hint: You'll need to buy a ticket for whole day, "Fritid" ticket is not for students.
Zones
If you just want to take the bus around the campus, around the city center, go to IKEA, to the supermarkets, etc, you'll only need to buy a ticket for zone A.
An exception is that if you want to go to the airport, you'll need to buy a ticket which includes zone B.
You can always check the zones here.