SEE: We see something static, unmoving. We also see the things around us, perhaps without paying much attention to them.
LOOK: We look at something in order to pay more attention to its appearance. We use look when we want to direct someone’s attention to something.
WATCH: We watch something when we want to know what happens (so therefore we generally watch something that is moving or that requires our attention).
One easy example to remember is by conjugating these verbs with the sky.
I see the sky.
(You're affirming that indeed the sky as a non moving object, exists and you're seeing it, but you're not going to fix your gaze at it as nothing in the sky is attracting your attention. You see it, and move on to do something else.)
I look at the sky.
(Now you saw the sky, but you're paying more attention to it. You're looking at it, fixating your view on something specific about the sky. Look, the sky is turning grey. Or look at the shining star in the sky. These sentences imply you actually have to hold your gaze at the sky to spot something that is in the sky.)
I watch the sky.
(Watching implies movement that is going to evolve into something or going somewhere. If I say, watch the sky, now I'm telling you to look at the sky, but hold your attention to the sky because something is about to happen that you need to 'watch.' If I had said look at the sky, then the action that I want you to look at, already happened and all you have to do is look at it. Watching requires a bit more attention, concentration, and that's why we probably say, I'll go watch a movie, since it requires attention. In this case, I see the movie screen in front of me, I look at it to watch the movie.)
Also, you can remember the difference by thinking of a baby.
If i tell you, see the baby on your right? All I'm saying is, spot the baby on your right with your vision.
Now look at the baby.
Looking at the baby implies that you will look at the baby and once you have seen it, you'll probably look at something else.
What if i say, 'watch the baby.'
This implies that now you'll have to hold your attention on the baby, or you'll have to look at it for a defined amount of time, which is not the same as "look at the baby."