Usually I knit adult sizes, because I know the bigger kids get fewer things. But this time I thought, just for fun, I'd make a smaller sock. Then someone asked me for a pattern for kids' socks, so I figured I'd just post here as I go. (If you need help with the basics, just click the link over there on the right for Elizabeth's Sock Class.)
Choose worsted weight yarn. Use needles a size or two smaller than you would use to make a sweater or hat; socks take a beating, and if they're more tightly knit, they last longer - and they're more comfortable to walk on.
Cast on 36 stitches and join into a circle. I use the twisted German cast-on because it is tough and stretchy; there are many YouTube videos to show you, if you want to try it. Whichever cast-on method you choose, make sure it will stretch easily over the ankle and heel when the kid is putting the socks on, and that it does not cut into the calf once it's on.
You have several options for the leg -- k2p2 rib, k1p1 rib, rib for 2 inches (please do at least that much ribbing) then switch to stockinette. . . you can add stripes, or knit the socks in a color block style where you do the leg, foot, and toe in different colors - as you wish. Make it pretty and fun or do the whole thing in navy blue -- there is room for every color option. The only thing that's not open to interpretation is that they must be strong and warm.
For this sock, I cast on with red, worked 8 rows of k1p1 ribbing, then switched to k3p1 ribbing when I changed to blue.
Don't worry about the uneven stitches where I put the sock down for a while -- when I wash this, everything will even out.
Work the leg until you have done 36 rows, counting from the cast-on edge. I'll tell you how to do the heel tomorrow.