Take a tram no.7 or 11 to Bergsjön. It´s a little lake 5 minutes walk away from the last stop. There´s no tourists, but many locals enjoying the lake and it´s surroundings. Water in the lake is really warm, because of it´s small size. Surrounding forests have loads of little paths which you can follow and enjoy the swedish nature. Also if you get there at the right time, you can pick blueberries. They are usually ready some time in July, it depends on the weather though.
The lake Surtesjön and Lärjedalen valley Nature reserve are both great, beautiful scenery north east of town. You just take a tram to Angered (end station) and then almost any bus to residential areas such as Rannebergen, Lövgärdet etc. and start walking. The city tourist office has trekking maps and can also tell you where to rent canoes etc. if you want to be out on the lake system. Just bring your own barbecue stuff as there are plenty of places where you can light a fire and camp in the forest but remember to respect possible fire prohibitions due to warm weather and to bring your waste with you.
Around Göteborg there is a lot of nice areas to explore by car or by foot. One is Mölnlycke, southwest of Göteborg, where you can find forests with lakes to swim and fish in the summer and forests to pick mushrooms in the autumn. Remember to check out how you are expected to behave while out in the nature and that you might have to get a permit to fish. The photo is from a lake in Mölnlycke at Midsommarafton, midsummers eve, a very Swedish tradition celebrating the longest day of the year and the beginning of summer.
One place people always miss is the beautiful scenery just outside the city. There is a beautiful forest/lake location that is called Delsjon...you really need a map for this. You can get there by bus and its a small hike, via a golf course to get there. There are two lakes and excellent places to have small barbeques. You can also go swimming in the lake if its sunny, but its better to swim where people can see. Another alternative is to hire a boat or kayak, there are some small clubs (Just follow the signs) where you can rent them for a few hours. There are no lack of lakes in Sweden and if your looking for some place to get away from it all, then get yourself a good map and go searching !!
Other places worth seeing: You can combine a typical tourist day by buying the Gothenburg card (95SEK) which you can use on all public transport, get discounts, free museum entrance and free boat trip to Älvsborgfestning ( a small island fortress icl. tour). There is a book with more inforamation about this at the tourist information offices.
There is a path between Hindas and Skatas.Take the tram 5 up tp Welandergarten and follow the road to Skatt.After a few houses the forest is the forest.