The problem
A circular needle can only knit a tube whose circumference is longer than the needle, and the shortest practical circular is 16" — a hat, not a sock. A sock is 7" or 8" around. Four solutions exist, and they all work; what differs is the feel in the hands, the gear, and where the join between needles shows.
Double-pointed needles
The traditional way: four or five short needles pointed at both ends. The stitches are divided over three or four of them in a triangle or square, and the spare needle knits each in turn. Crystal Palace bamboo DPNs come in sets of five at 8" long, from US 1 up.
What it is like. Fiddly for the first inch, then natural. Nothing to buy beyond the set. The sock is in the round from the start, the heel and toe are easy to see, and the fabric is handled less than with a loop of cable. Bamboo DPNs grip the stitches so the idle needles do not fall out, which is why they are the beginner's set; metal DPNs are faster and slipperier.
The catch. Ladders — a loose column of stitches where one needle hands over to the next — until you learn to snug the first stitch on each needle. Five needles is a lot to drop on a bus. And a needle left sticking out of a bag will find a hand.
Magic loop
One long circular, 32" to 40", with the cable pulled out in a loop at the halfway point so that half the stitches sit on each tip. Knit across one half, pull the cable through, turn, knit the other. addi's 40" Rockets and Turbos are the usual choice; a supple cable matters more here than anywhere.
What it is like. Two needle tips, nothing to drop, the whole sock on one needle, and both socks at once if you want (two-at-a-time is a magic-loop technique). The same needle knits a hat, a sleeve and a mitten, so it is the fewest needles to own. Most knitters who learn socks today learn this way.
The catch. The cable is pulled through every half round, which some hands dislike, and the two join points can ladder just as DPNs do until the first stitch on each side is snugged. A stiff or kinked cable makes it miserable; a good one makes it disappear.
Two circulars
Two 24" circulars, half the stitches on each, each half knit with its own needle's two tips. The same idea as magic loop without pulling cable through: pick up the needle whose stitches are next, knit them, drop it, pick up the other.
What it is like. The gentlest on the hands of the four — nothing is pulled, nothing is cramped — and the join points are stable. Also fine for two socks at once.
The catch. Two needles to buy in the same size, and in the first rounds it is easy to knit one half onto the wrong needle. Mark one of the two.
Short bent circulars: addi FlexiFlips
addi's answer to all of the above: a pair of 8" needles, each with a short cable in the middle and a tip at each end, used like double-points but with two needles holding the stitches and one knitting. In stock from US 0 to US 8; the addi guide sets them beside the rest of the range.
What it is like. The speed of DPNs without the dropped needles, only two joins instead of three or four, and the needle lies flat in a bag. Knitters who found DPNs fiddly and magic loop fussy often settle here.
The catch. The bend takes a round or two to get used to, the pair does one sock at a time, and they are the one sock needle that is no use for anything wider than a sleeve.
Sizes for socks
A standard fingering sock knits on a US 1 or US 1.5 (2.25 to 2.5 mm) for a dense fabric that wears; a US 0 for a tighter one or for colorwork; a US 2 for a looser sock in a plumper yarn. Sport-weight boot socks take a US 2 to 3. The needle size chart has the millimeters, and the note that addi's US 1 is 2.5 mm while Crystal Palace's is 2.25 — match the pattern's millimeters, not the number. The sock yarn guide covers the yarn side, and a 100 g skein from the sock shelf is a pair.
Ladders, whichever you choose
Ladders are not the needles' fault. They come from the first stitch on a new needle being worked loosely, because the previous needle's last stitch pulls away from it. Three cures, any of which works: knit the first two stitches on each needle a little tighter; shift the division point by a stitch or two every few rounds so no column is always at a join; or knit one extra stitch from the next needle before changing, so the joins travel round the sock. After a few pairs you will stop noticing.
Which to start with
If you own a long circular, try magic loop first — nothing to buy, and the skill carries over to sleeves and mittens. If you would rather hold needles than cable, a set of Crystal Palace bamboo DPNs in US 1. And if you have tried both and liked neither, FlexiFlips exist for exactly that knitter.