See is the ordinary verb to say that something ‘comes to our eyes’. It doesn’t necessarily mean that we are paying attention.

I saw Joe yesterday.
Suddenly I saw something strange.

Progressive forms of see are not normally used with this meaning.

I can see an elephant. (NOT I am seeing a elephant.)

Look (at)

When we look at something we are trying to see what it is – we are paying attention. Note that we can see something even if we don’t want to, but we can only look at something deliberately.

He looked at the baby with his eyes full of love. (NOT He saw the baby with his eyes full of love.)
She looked at the picture.
‘Look here.’

Note that look is followed by a preposition when there is an object. When there is no object, there is no preposition.

Look at the photo. (NOT Look the photo.)
Look here. (NOT Look at here.)

Watch

Watch has more or less the same meaning as look. We usually use watch to talk about looking at events that change or develop.

Did you watch the football match yesterday. (NOT Did you look at the football match yesterday.)

What are you doing? I am watching the TV serial.

Watch him – I am certain he is up to something.

Watch is normally used with TV.

Don’t spend too much time watching TV.

Both watch and see can be used to talk about films and TV programs.

We watched/saw a great film yesterday.

See can be followed by if and weather. Look and watch are not normally followed by if or whether.

Let’s see whether she is in. (NOT Let’s watch/look whether she is in.)
See if there is any food left. (NOT Look/watch if there is any food left.)