Summertime I suggest you take one of these old trams that gives you transport from Drottningtorget(Queens square) to Liseberg. This is ringlinjens tram museum that offers tours in old wagons from 1902 to 1969. The tram showed on this picture is a mustang from 1948-49. The tram is rebuilt for right-hand traffic in 1967 and by then got the doors on right side.
A service I recommend is using resrobot(travel robot) you see here>>>. With this one you just write the goal and you'll then get the quickest way going somewhere in Sweden.
A good idea could be buying a touristcard for Gothenburg that gives you possibility to travel all around the city and archipelago for 85 skr(about 8€ or 9$). You'll get all these cards in vasttrafiks shops like tidpunkten.
You are also able to get an application for your mobile where you are able to see what bus, tram or boat you use at the moment to getting you anywhere in Gothenburg. You'll see travel planner for most devices on https://www.vasttrafik.se/#!/en/travel-information/4/
Another wonderful app is the touring guide who tells you everything about the places you are passing. Get it here>>>
Airports in Gothenburg
At the moment we just got one.
Bus in Gothenburg
Another affordable way to get around in Gothenburg