Is it correct that he wanted to find an alternative route to India?
So when he did reach America, did he think he was in India because I've also heard that he thought he had reached Japan? Why was "Indian" coined as a term for Native Americans, if he thought he was in Japan?
More can be said here, but this answer might be of interest for you.
It's pretty much what you said. He wanted to find a trade route to India.
To get to India from Europe you had to sail down the west coast of Africa, round What is now South Africa, than up the east coast of Africa, than past the Middle East and finally in to India.
Columbus thought that you could sail west and reach the east coast of India everyone at the time known this was impossible and without American Columbus would have died.
At first he definitely thought he was in India as he called the land the West Indies and the people the Indians. But it seems at one point he decided he was In Japan or China. After awhile he figured out that he had found a new landmass quickly abandoned his India trade route idea and began to focus on the new world.