Taking the Västtåg between the central station and Gamlestaden would be very practical for me. Do you have to get a ticket for that?
Yes, you need a ticket. Public transportation in Gothenburg is not free. You can purchase a ticket from a machine at the station, or a manned ticket booth. The machines only take credit/debit cards. A ticket for an adult costs 30 SEK and is valid for 90 minutes on all Västtrafik lines (so train, tram, and bus).