What could one study to become involved with an anime production company or what should I study to become an animator? I have always been interested in drawing scenes and storyboarding.

Reading/watching Richard Williams' "The Animator's Survival Kit" is a must, every animator out there has gone through it. There are several books on storyboarding (I have "Exploring Storyboarding" by Wendy Tuminello) which are helpful. To become a decent animator, you should also master everything that concerns drawing like the fundamentals, anatomy and perspective.

I would highly recommend going through Preston Blair's 'Animation' before approaching an Animator's Survival Kit as it has a much better structure for beginners. You will learn more info, more quickly using Preston Blair than using Animator's Survival Kit first, but should definitely move onto that Animator's Survival Kit afterwards.

Some good resources:
Preston Blair 'Animation'
http://animationresources.org/instruction-preston-blairs-advanced-animation/

Brian Lemay Animation. Useful animation exercises. The video demonstrations are also helpful (I usually watch them at 1.5 times the normal for quicker consumption)
http://www.brianlemay.com/Pages/animationschool/animationschool.html

Animation Physics. A good resource for *understanding* animated movements.
https://www.youtube.com/user/AlejandroLuisGarcia/playlists

'Figure Drawing for All It's Worth' - Andrew Loomis. Not Animation. But covers how to draw the figure in correct perspective without reference.

One of the few tutorials on 'background animation' I've seen on the web
https://www.youtube.com/watch?v=TIrEhOP5aiE

Storyboarding.

Framed Ink by Marcos Mateu-Mestre
Directing the Story by Francis Glebas