Gothenburg is a beautiful city, and a bit more chilled out than Stockholm- though I love them both. Maybe it’s something to do with being surrounded by water, but there’s something really calm about both cities. I’ve got a good friend who used to live there (now moved to Hammarö – another area worth visiting!), so been there several times.
There’s plenty to do as a tourist, but obviously if you’re going to settle there you can discover it in your own time. One of the nicest things you could do as a family, to get a feel of it, is take a Paddan boat tour from Kungsportsplatsen (across from the Tourist Office). It’s a good way to experience the city’s beauty; some of the bridges on the Gota canal are so low, everyone has to lie flat whilst going under them. We loved it.
It’s definitely worth visiting the ‘Fish Church’ (Feskekörka) – a huge place, architecturally designed like a Church (and actually consecrated too, I believe!), which is a big fish market. Fantastic for shopping for fresh seafood.
The Haga district of the city is very bohemian, and worth just mooching around – looking at beautiful little shops and stopping for the traditional Fika. Obviously, you got to have a cinnamon bun (Kanelbullar) with your coffee. There are lots of little lovely places to eat around there, as well – home-made meatballs are great, but they do great stuff with Reindeer meat too, not to mention fresh seafood.
If you’re into outdoor swimming, don’t miss visiting the Delsjön lake: https://www.goteborg.com/en/delsjon/ - a heavenly place to swim outdoors in the summer. In fact, swimming outdoors is one of the most life-affirming things you could do in Sweden. It’s so clean, and geared up for it – with bathing stations/steps running into the water, and often lovely little cafes nearby. If you do this in Midsummer… well, it’s just blissful.
Speaking of Midsummer – not sure if you’ll be there for it? But the cities tend to empty, and everyone moves to the countryside for the weekend (it’s a national holiday). Many Swedes got little huts in the country, so make some friends and celebrate like a Swede. We’ve been to one at our friend’s Midsommar Garden party a couple years ago, and it’s the best party I’ve ever been to.
Other great attractions are the Liseberg amusement park – my husband shed about 30 years when we were there – he’s a nutcase for that sort of stuff. Even if you’re not into rides, the place is amazing: https://www.goteborg.com/en/liseberg/
The Botaniska trädgården (Botanical Gardens) are a must as well. I think it is the biggest one in Sweden. There’s a famous Volvo museum too, but I have never been.
As for the Systembolaget (or simply System, as they call it) – as @peterm said – it’s like entering a large and very well organised drinks supermarket now, rather than the old slightly Soviet system. There’s always a good selection at fairly good prices too; the only thing to be aware of are the opening hours. A few times we forgot about the closing time, and ended up having to buy very weak lager from the supermarket (I think they can only sell alcohol up to 3.5% in supermarkets – but might be wrong).
To be fair, my friends always have wine in the house, so it only happened when we travelled on our own. If you stock up – it’s not a problem. Interestingly, the Swedes love their wine boxes. I think a significant majority buys wine this way.
If I can think of anything else I’ll let you know…