I wonder how Hollywood became the place were movies are made.
Hollywood began as a neighborhood in 1870. 
And when movies began to be produced there?
I am not sure the exact date but in 1910 it was consolidated  with LA. 
Any idea what was the first studio to open in Hollywood?
Good question.  I have no idea.  But classic cinema developed from there between 1917 though 1960. 
And are most movies still produced in Hollywood?
I highly doubt it since film production companies are established all over the world. 
What other places are famous for producing movies?
Well there is the Cinema of the Philippines and the Cinema of the United Kingdom.