Most people learning to cast get into trouble by working it too hard and by letting the rod tip come back too far behind them. (over the shoulder)

Casting a fly is more like snapping a whip.
Casting bait is more like throwing a rock.
They are not the same type of action.

With a fly line the weight of the fly is nothing and you are really just casting the line. The fly actually slows the line down as you cast. Flies have a lot of wind resistance and no weight.
You can even practice casting with no fly at all, but it is hard to see where you are casting then and the line by itself travels farther than it does when it is dragging the fly through the air.
I used to tie a piece of yarn on my line for practicing. You don't get the hook sticking into things then or lose nice flies when you manage to pop them off

If you think of a vertical clock face your rod tip should move from about ten or 11 o'clock to about 1 o'clock.
Use your elbow to move it instead of your wrist. The rod and your forearm should stay in a straight line as you move them.

After that it is just a bit of timing. You need to pause just long enough between the forward and back motions to let the line mostly straighten out
If you don't pause long enough on the back cast the line does not get out flat enough behind you to cast well.
You also start popping the flies off the leader because you are snapping the line like a whip. Sometimes you even hear them. You don't want the fly to snap like the popper on a bullwhip.
If you wait too long on the back cast it starts to sink. You will get flies snagging the grass and you get a weak and loopy forward cast. when you have a big vertical loop rolling forward on your cast you are waiting to long between you back cast and forward cast.

It really helps to have somebody watch you from the side when you practice. They might even have to hold your arm at the right position because most of us are really bad at telling how far back over our shoulder and down we are letting the rod tip drop.

I wish I knew a good book or video to tell you.
I don't know one though.
I was taught to cast by an old fellow years ago and the biggest things he kept on about was not moving the rod tip to far forward or back and to try for accuracy instead of distance at first.

If there is a sportsman's show near you they often have casting clinics for beginning fly fishermen. They can be a great help to get you started. Maybe if you have the wife or kids in tow they can be shown the form of a cast to watch for and can help you practice by telling you if you are moving the tip too far back.