Cast 60 stitches onto your circular needles. Yes, it looks far too small right now, but have I ever steered you wrong before? Now comes the fun part, 'knitting in the round'. Yup, you're going to be going around in circles. Before that, straighten out all your stitches & make sure that they are all pointing inwards (or down, if that's your thing). You don't want to get your stitches twisted up, or you'll end up knitting a Mobius strip. A surface with only one side it may be, a hat it isn't. You'll need a stitch marker (you can buy stitch markers, and they come in different colours & styles. For this I'm using an Ankh ring sent to me by the lovely Cynthia as my stitch marker. Just slip it onto the needle next to the last cast on stitch. The stitch marker is just there to show you where the end of the row is. When you knit your row & get to it, you just slip it from one needle to the other & carry on knitting. It doesn't get knitted into the hat, just goes round & round the needles.
Another magic power of the circular needle is it turns a Garter stitch into a Stockinette. Yes, hot snow falls upwards. Why does it do this? Because you're knitting in a spiral, rather than the back & forth of regular needles. So if you want to make a garter stitch hat, you'll need to knit 1 round, purl 1 round & keep repeating. Or you can knit an inside-out, garter-that-becomes-stockinette magic-hat instead.
Keep knitting until you have 6" or 7" or so of knitting (a little less if you're making a Beanie style hat, a little more if you want a big chunky brim). Now comes the decreasing. Just relax. get yourself a cup of tea. Or a glass of wine. Some Malbec would go down well. Knit 8 stitches, then knit 2 stitches together. Repeat until the end of the row. Now knit 7 stitches, and knit 2 together, and repeat until the end of the row. Now knit 7 stitches, and knit 2 together, repeat to the end of the row. At this point you & the circular needles will have to part ways. yes, it's been a whirlwind romance, but you'll meet again soon. Time to meet the dpns, the double pointed needles -discovered by a the Mad Arab Abdul Alhazred in the ruins of Babylon & used to knit a cabled tote bag for his early draft of the Necronomicon. Wield them and despair!
Okay, so they're not that bad. But look, they make an Anarchy symbol when used to knit a hat. Coincidence? Probably, yeah.
You can just slip your stitches onto 3 of the dpns, spread them out equally & continue knitting the rounds of decreases. Or you can knit straight onto them, whatever works best for you. With the 4th needle knit off the dpns (it sounds complicated, but it isn't. Just take your empty needle in your right hand & the dpn with the stitches you need to decrease next in your left. Knit & decrease from left to right, and use the now empty left hand dpn to do the same again to the next one). Knit & decrease for as long as you can stand to (I keep going until I have 3 or 4 stitches, but I'm Peculiar), cut your yarn, leaving a tail of several inches, and thread through the remaining stitches. Tie off & weave in any loose yarn left (don't forget the bit at the brim where you cast on). Turn it right way around (if you've been knitting it inside out)
There you have it. A very fine hat.