One of the simplest things that you can do with your kids are knitting blanket squares. It is also a great project if there is more than one kid.
To make blanket squares you need lots of wool. I recommend that you use the same brand and ply when your child is just learning as he or she will have enough trouble getting a consistent tension without trying to knit with wools of different textures and thicknesses.
I also suggest that you pick up several different colors so that your child does not get bored knitting a lot of squares that are the same.
8 or 12 ply wool tends to be the best choice as the yarn is not too fine for your child to work with and the larger plys tend to be more forgiving of mistakes.
For 8 ply wool, start with a pair of 4.00mm needles which are not too long.
The longer the needles the harder it will be for your child to manage them!
For 12 ply wool you can use 5.50 mm needles.
(If your needles don't come in metric sizes check out the page on knitting needle sizes to find the correct size.)
On the band of your wool you will find the recommended tension for that particular wool. For example the recommended tension for the 8 ply wool I am using at the moment is 22 stitches and 30 rows for a 10 x 10cm square.
All you need to do to work out your number of stitches and rows is to decide how big you want your squares to be and then do the math.
For example if you wanted squares that are 20cm x 20cm just double the number of stitches and rows specified on the band.
Your child's square is not likely to be 20cm x 20cm exactly, but that doesn't really matter as long as they all end up the same size!
If you want squares that are 15cm x 15cm multiply the number of stitiches and rows by 1.5.
To start your child's first square cast on the required number of stitches and they can start knitting.
If you keep a close eye on his or her progress you can double check the number of rows by using a tape measure and adjust the numbers of rows accordingly.
Most people start their kid's knitting in garter stitch as it is the easiest stitch to do - just knit every row. But kids being kids, sooner or later they will get sick of doing garter stitch even if it is in different colors!
When you get to this point you can introduce a few other stitches. Show your child how to do purl stitches and they can make some squares in stocking stitch as well.
Then you can introduce color changes - stripes - or textured patterns like four rows of garter stitch followed by four rows of stocking stitch.
As your child's knitting skills improve you can move onto more complicated pattern stitches like stocking stitch and garter stitch checks.
Once you and your child have finished knitting blanket squares, it is time to start sewing or crocheting them together. Usually what I do is to crochet around each square in a single colour and then hand sew them together.
The crocheting provides a border effect and I think it finishes the blanket off nicely. It also helps if not all the squares are quite the same size. Once I have them all blocked together I used the same wool to crochet an edge around the whole blanket.