the basic beanie
There are many many patterns /instructions/ tutorials out there for beanies. By definition (mine) a beanie is a simple hat which fits closely on the head, and generally covers the ears. Once you look at the patterns in detail, however, it soon becomes evident that there is really one pattern.
 Make a flattish round piece for the crown, then go straight until the hat is big enough. Stop.
Well perhaps a few more details, but really there are not a hundred ways to cover the top part of a head with a plain hat, or at least there need not be a hundred different sets of instructions.
First you make somewhere to crochet into, a  magic ring, several chains slip-stitched into a ring or even a single chain. Then you crochet into that. I have seen instructions ( and followed most of them) for anywhere from 3 to 14 stitches for this first round. Why such variation? Sometimes it seems, just to be different from the last pattern! Basically though the taller the stitch you want to use, the more stitches you need to start with. So for sc (single crochet) you will probably start with between 3 and 8, for hdc (half double crochet) 7 to 11 sounds ok, and for dc (double crochet)  10 to 14 stitches into the initial ring will work.
The next step is increasing, so that the rounds will lay almost flat. It is very easy to figure this out. Increase each row by the number you started with. So if you first used 6 sc, the next round will end with 12 sc  ( so  2 sci n each sc). Then for round 3 you want another 6, to end with 18sc ( sc in first sc, 2 sc in next sc, repeat). And the fourth round will increase by another 6 to 24sc (sc in first 2 sc, 2sc in next sc, rep). And so on until you have a round which is big enough. For a real shorthand version of the instructions I write R1-6, R2-12, R3-18, R4-24, R5-30, and so on .
A hat with hdc would work up for example as R1-11hdc, R2-22hdc, R3-33hdc, R4-44hdc...
And for dc, R1-14dc, R2-28dc, R3-42dc, R4-56dc.
And how big is "big enough "for the flat part. Depends on final hat size. Child size hats usually recommend 13.5cm (5 inches) adult hats 15- 16.5 cm (6-7inches). But there is a better way to be sure. Measure the circumference of the head in question, I just checked an available head and it is 57cm (22.5in). since a hat generally has "negative ease" to keep it from blowing away in the wind, 55cm (or 21.5 in) should be good.  To find Diameter, from Circumference  [D=C/pi] 55/3.14=17.5cm, or 6.9 inches.
Once you reach the correct size for the crown, stop increasing and just continue one stitch in each stitch until it is big enough. Again checking the head is best but it will vary from 12 to 18 cm depending on the head ( 5 to 7 inches) from the end of the increase (ie the edge of the crown)
There are 2 basic ways to go around and around. Either crochet in a continuous spiral, or join each round with a slip stitch and ch 1 to 3 (sc to dc) to begin next round. Spiral is fine if you do not plan to change colours along the way, but join and chain will give smoother colour changes.
Here are a few sample beanies showing the possible variations due to yarn size.

very old wool, probably  size 1, crochet with 2.5mm hk. Starting round 8sc, 53 sc rounds. Fits that head I mentioned above.

cotton yarn size 4, 5mm hk. 14sc in ring then dc for crown, hdc on sides. Total 17 rounds. To fit medium child.

acrylic/rayon yarn, size 5, hk 8mm and 6.5 for edge, 12 dc in ring, 10 rounds of dc then 4 rounds blo sc. fits adult.

There is now a system of sizing yarn. It is not perfect, but if you have access to an old stash with no sizing at all, you will see how much better it is to at least have a starting point!