Where would one start doing 3D in the web?

I need to do some 3D in the browser. I've got no idea where to start. Wouldn't want to lose too much time looking for an appropriate solution.

For example I have the following picture in my mind - an area with ground and no boundaries one can move in in all directions, using keyboard and mouse, including up and down, filled with seaweed like plants growing up from the floor slightly swaying, joints on them should be interactive.

Would be grateful if you could orient me in the right direction.