Watch is a little like Look (at) but implies that something is happening or is going to happen. We watch things that change, move or develop.
I watched a baseball game last weekend at Tokyo Dome.
I enjoy watching TV in the evening after work.

Here are a few two part verbs and idioms that use watch.

watch out
To be careful or on the alert; take care.
Watch out for cars when you are crossing the street.

watch over
To be in charge of; superintend.
When I was young my big brother watched over me. He didn't like babysitting though.

Idioms:
watch it
To be careful:
I had to watch it when I stepped onto the ice. I hadn't been iceskating for years and was afraid I would fall.

watch (one's) step
1 To act or proceed with care and caution.
Watch your step on going up the stairs! They are slippery after the rain.
2 To behave as is demanded, required, or appropriate.
You should watch your step around the new boss. I hear he is very strict.