In The Dead Poets Society, how is romanticism and transcendentalism showed? I have the nature, and intuition, but am not sure about anything else.

First thing to do is understand romanticism and transcendentalism.

Transcendentalism was an idea that basically says you need to do what your own conscience tells you is right.  Never mind what society tells you, just do what you believe in.

In this context, the main point of Romanticism is that emotion is superior to logic and intellect.

With that in mind, you should be able to see these things in the movie.

Keating encourages the boys to see things from their own perspective.  He encourages them to climb up on their desks, to go outside the classroom, to look at things differently.  He encourages that one boy to act in the play, no matter what his dad says.

When the boys rip the boring essay out of the book, they are approaching poetry from an emotional point of view rather than from the essay's more logical point of view.  When the one boy writes the love poem to the girl he is going on emotion.

The movie shows people being encouraged to do their own thing and to have feelings rather than just logic.  These things are right in line with Transcendentalism and Romanticism.