Seurasaari is a beautiful little island easy to get to for nature, walks and beaches. There is a bus (number 24 e.g. from Lasipalatsi going west) that takes you to the white bridge you walk over to the island. During the summer and some events there are cosy little cafés, and there is at least one bbq for everybody to use. Seurasaari also hosts on Open Air Museum showing the traditional Finnish way of life in buildings.
Seurasaari is a good place to go to explore the nature with kids. There are some swings around too but in my opinion Seurasaari is more about the nature. My kids loved feeding the relatively tame squirrels and small birds from their hands, and they also love the church boats (huge rowing boats that were traditionally used to get people to the churches that in lake areas were often located on islands) as in their opinion the Hattifatteners come by those boats. The ice cream kiosk in the beginning of the bridge is also an important stop.