Please check the PDF overview drawing attached. The spruce wood I have used is quite massive (can even bear the load of an 65kg adult), so for kids only you could also use thinner wood. Check what you can get in your local supply store.
After cutting all parts according to the drawing I have chamfered all corners (to avoid injuries), grinded the wood and then pre-assembled the table with screws to check fit. Check out the pictures to see how many screws I have used.
Then I have disassembled the table again to paint all the parts in white color (primer). Before coloring each part individually I have grinded the wood slightly again for a fine surface. If you like my colors check the PDF drawing for the color instructions (in German only, sorry).
As I wanted to see the wood structure after painting I have mixed the water-based color with quite a lot of water (about 1:1) and applied this “colorful” watermix with a household spoon on the wood surfaces. The advantage is that you apply the color very quick and homogenous, save color and it dries very fast. Take care you protect the shop floor as colorful dripping cannot be avoided.
So then after drying assemble the table and let the kids enjoy. Make sure the kids cannot insure themselves by e.g. screws heads or screws which are too long.
If you feel like: drill a hole in the center board for a sunshade.