I wrote up the following list for a friend with a young daughter who wanted to knit. It is a list of basic techniques, the later ones building on the first ones, along with hints about how to combine them to knit projects. It depends on a number of videos from a yarn website, and is geared for continental knitters (knitters who hold the yarn in the left hand and pull it through the loop with a picking motion). But the website also has videos for English knitting, where you throw the yarn around the needle and pull it through.
how to knit continental-style
weaving in tails in garter stitch
With these you can make a doll's scarf or, with more yarn, a child's scarf in garter stitch which has a bumpy surface. (Cast on as many stitches as you wish for width, knit back and forth as many rows as you wish for length, then bind off.)
knit two together (k2tog)
YO increase (yarn over)
With this, a ball of cotton yarn, and size 7 U.S. needles, you can make pretty dish cloths. Or you can just make plain ones in garter stitch like the scarf, by casting on 44 stitches and knitting until you get a square.
Purling allows you to do rib stitch.
knit one purl one rib stitch (K1P1)
With this you can make a ribbed doll's scarf. (Cast on about a quarter more stitches than you think you need, using an even number; knit one purl one repeating until the end of the row, turn work and purl one knit one; continue until the scarf is long enough and bind off. Then try a scarf in knit two, purl two ribbing.)
knitting in the round (stockinette) with double point needles
long tail cast-on (allows cloth to stretch)
weaving in tails in stockinette stitch
With this and some 10 inch long double point needles, you can make a hat. (Pattern in The Knitter's Handy Book of Patterns at the library.)
M1 increase (make one)
With these, some 5 or 6 inch long double point needles, and some stitch markers, you can make a pair of child's mittens. (Pattern in The Knitter's Handy Book of Patterns at the library.)
stockinette stitch on straight needles
mattress stitch for sewing pieces together
With this, you also have all the techniques needed for a sweater or vest. Try The Knitter's Handy Book of Sweater Patterns at the library, or Knitting without Tears.
After that you can get into things like lace (yarn overs, decreases), cables (crossing stitches), garment shaping (short rows, decreases and increases), dropped stitches, and stitch patterns.