One needle
A US 8 (5 mm) bamboo circular, 24" to 26" long. Every part of that is chosen.
US 8 is the size that matches worsted-weight yarn, which is the weight to learn on: big enough to see each stitch, small enough to make a fabric rather than a net. Bamboo grips the yarn a little, so stitches stay on the needle while your hands learn where they go, and it is warm and light (the material guide compares it with metal and wood). A circular rather than a pair of straights because it knits flat exactly as straights do and also knits in the round, so the first hat does not need a second purchase; and because the weight of the knitting rests on the cord in your lap. 24" to 26" is long enough for a scarf, a dishcloth, a hat or a sweater piece and short enough not to tangle.
Crystal Palace's 26" bamboo circular in US 8 is the one we would hand you. If you have been told to learn on straights, their 12" bamboo straights in US 8 are the same idea; the circular vs straight guide explains why most people move to circulars anyway.
One yarn
A light, solid-colored, smooth worsted wool. Light because you can see the stitches; solid because a variegated yarn hides mistakes you need to see; smooth because a fuzzy yarn hides them too, and a slippery one falls off the needle; wool because it is forgiving, springs back when you pull it out, and is pleasant to hold. One 100 g skein is a scarf or a hat with yarn left over for a second try.
Brown Sheep Nature Spun Worsted is the classic learning yarn — plied, smooth, a huge range of solid colors, inexpensive — and Plymouth Worsted Merino Superwash is the softer, machine-washable version for someone whose first project is a gift. Both are on the worsted shelf. Avoid cotton (no stretch, hard on the hands), acrylic (squeaks and does not block), and anything with mohair or a halo until the second project.
Three small things
A needle gauge, the card with holes that tells you what size an unmarked needle is, since you will own unmarked needles sooner than you think. A tapestry needle for weaving in the ends — the blunt kind, with a big eye. And a handful of stitch markers, little rings that sit on the needle and tell you where a hat's decreases begin. Point protectors, which stop stitches sliding off the needle in a bag, are the fourth thing, and the accessories shelf has all of them. Scissors you already own.
What not to buy yet
A set of every size, an interchangeable kit, double-pointed needles, a cable needle, a row counter, a swift and ball winder, and more than one skein. Each of those is for a project you have not met yet, and the project will tell you which one to buy. The one exception is a second skein in the same color if you already know the first project is a scarf — the hat and scarf yardage guide says why.
The first project
A garter-stitch scarf teaches the knit stitch and nothing else, which is the point. A dishcloth teaches it faster in cotton, at the price of sore hands. A simple hat in the round on the same circular, once the scarf is done, teaches purling, ribbing and decreasing in one small piece with a result you will wear. Our pattern explorer has free patterns for all three.
When the needle is wrong
If the fabric is stiff and the stitches hard to move, go up a size; if it is loose and open, go down one. The yarn weights guide has the table of which size goes with which yarn, and the needle size chart the millimeters. A second needle in US 7 or US 9 is the cheapest fix for a first project that is not going well.