We spent 5 weeks around holidays in cold Midwest and I made Kitten these slippers.
They are very easy to do and are knitted in one piece each. Basically you are making a T-shape. Start from the horizontal bar. Measure your child's leg from the point where you would like the slipper to stop, around the heel and back up to that point again. Determine how many stitches it will need in your yarn (or just cast on, knit several rows and try it on like I did). Continue working in whatever stitch you like, mine is garter stitch until your piece is long enough to wrap around child's ankle. After that measure around child's sole and determine how many stitches you need to cast off on each side. Continue until you reach the sole's length and a little extra to go around the heel. Decrease by knitting every 2 stitches together every other row until you have less than 10 left. Break off yarn leaving a long tail, pull it through remaining stitches and stitch until you reach your cast off row (if you would like for it to look more like a sock keep going to the top). Sew the back together.
Now, if you want these to be slippers and not socks, you need to make soles from leather or suede or they will be very slippery. I traced Kitten's foot with the slipper on and just did a mirror image for the other foot. I cut soles from suede and pre-made holes because it was pretty thick. After that I just stitched them on trying to keep them on the opposite side from the opening, which was trickier than you may think!
The idea came from All Year Round (Lifeways) book but I modified all the directions and came up with a more generalized way, so I thought I'd share :)