A lot of how you make a béchamel is technique -- here's how I learned (from my italian great grandmother).
You'll need a wooden spatula for stirring, or a wooden spoon if you don't have the spatula.
- Melt some butter (exact amount depends on how much thickening power you're trying to get, I'd typically use 2-3 TB), and let it foam a little bit, but not brown.
- Sprinkle on as much flour as you had butter. (you can use more, but not more than about twice as much flour to butter)
- Stir in the flour, and it'll form a clay-like lump. Spread it across the bottom of the pan to cook.
- Let it cook for a minute or two, redistributing it a couple of times during the process. Exact cooking time depends on how hot the stove is; cook until it loses the kinda greasy look. (the longer you cook it, the less thickening power you'll have, and it'll start to take on a nutty flavor, which you actually do not want for this; you don't want it to pick up color; you should stop before it hits a straw color, which is enough to cook out the raw flour taste).
- Add a little milk (it can be cold, that really doesn't matter, but you only want to add a couple of TB.
- Stir thoroughly, and it'll start to look like paste.
- Add some more milk. Maybe a little more than the first addition.
- Stir thoroughly. it'll look like a thick batter, or maybe runny paste.
- Keep repeating the milk/stirring 'til you get the consistency you want. You never want to add more milk than what's in the pan so far (and it's better if you only add about 1/2 as much volume as what's in the pan; the slower you add the milk, the less stirring is required to mix it back in).
- Add a pinch of salt (but only if you use unsalted butter), and grate a little nutmeg over it, and stir it in.
You can hold it if you keep it just below a simmer. You'll need to stir it once in a while, to keep it from burning on the bottom. If it's getting too thick, just add a little more milk.
If you heat it too high, you'll actually loosen it back up. You want it at most a low simmer. It'll also thicken up some more as it cools, so you might want it a touch runnier than you want to serve it at.
If you're using this for an alfredo or mac & cheese, the pasta will absorb a lot of liquid, so you'll want it fairly loose.