Its been a couple years since I've been, but...
-Get a Goteborg CityPass as long as it isn't a big budget issue. Comes with public transport and entry to pretty much all the major museums, gardens, and entry to the amusement park (though no ride wristband).
-If your schedule allows, get on a boat and take a ride to one of the islands in the archipelago. There's several of them and boats in summer leave quite frequently. I went in September, and they much more rarely went.
-Canal rides are fun there. There's a bridge with hilariously low clearance before you end up in the main harbor.
-City Museum of Gothenburg is quite nice.
-Trädgårdsföreningen is a nice park space. After 4 or 5 PM it becomes free to enter IIRC, should you be on a really tight budget. Of course, some of the things inside like the conservatory will be closed at that time,
-Universeum is a pretty swell science museum/center sorta place but I can see it not being a huge value to some. Same with the amusement park next door (Liseberg) which is also very good for what it is.
-There's a building along the harborfront nicknamed the lipstick building. It's pretty obvious which one it is. They do have an observation deck you can poke around.
-The Martitime Museum in the harbor is quite good. If you have an interest in boats or military history, its worth a perusal.
-I stayed in downtown at the Hotel Avalon when I was there, which is pretty deluxe and cushy, and that meant I was very conveniently located to the food market. I could watch people in the café from my stepout balcony. Anthony Bourdain can be a pretentious dick at times, but he's not wrong about the value of the central urban food market. It is fairly recently renovated. I know, because they were doing the work when I was there in 2010!
I hope that helps.
"It's not some safe thing like Fugazi where everyone sits down and eats their tofu and goes 'wow man, that's revolutionary' " - Jerry A of Poison Idea