This is one of the biggest parks in Moscow near the city center, on the North East side. We can highly recommend to discover the park by Velobike (see "to organize-practicalities), which we rented at Sokolniki metro station. This is the perfect way to get quickly beyond the crowded amusement and food truck part which is clustered at the beginning of the park.
On the left side of the park you'll find the major kids activities, from amusement, to climbing, to dance floors and even an astronomical garden. There's even a cart center in the exhibition center. If you bike a bit further and central on the unpaved paths you will discover nice, quiet and scenic areas where you can relax.
There are many concerts and exhibitions, so check out their website for the program.
Photo credits: Martin du Plantier & Sokolniki website