If you want a walk

If you want a longer walk and to see some of the nature nearby, we have Änggården/Änggårdsbergen, the area just behind the Botanical Gardens. And Delsjön, which you can take a tram some of the way to the stop Göteborg Bögatan, then walk up the hill to the lake. Here you can take a loop around stora delsjön, or both stora and lilla delsjön. There's an app called naturkartan which might be useful for maps with walking trails on (but not entirely sure if it is available in english...), otherwise allTrails or something similar.

You can also take the tram (number 11) out to Saltholmen, then take a ferry out to one of the islands in the southern archipelago, like Styrsö or Brännö. You could buy a day pass for the tram(which covers all ferries too), then take boats between the islands too.

For a shorter walk there's the Botanical Gardens, the big park Slottsskogen, or Trädgårdsföreningen, which is right next to the central station.

Other...

You can wander around town, nice cafes and fika opportunities in Haga ( Haga Nygata), Linnéstaden (Linnegatan). If the weather is nice, sit outside and have a beer somewhere like Stora teatern, or in Trädgårdsföreningen.

There's several museums if that's your thing, and Liseberg is open most days too (I think) if you want a little theme park action.

Also there's discgolf or frisbee golf in Slottsskogen, can be fun or very frustrating.

For the climbers

There are a couple spots, but mainly from town you can take the tram/bus out to climb at Utby/Fjällbo. Topos available at gryttr We have one pad you can borrow, and others can be hired from our local wall, Backa Boulder. Let me know if you're keen I can provide more info/help.