First off, if you're asking NYC filmmakers for help, don't say "LA is better..." Most of us disagree with you.

Secondly, the websites mentioned above are great. Here's my breakdown.

For experience - craigslist
if you already have some experience - Mandy.com
If you have some experience and know what you're doing - StaffMeUp.com

Facebook groups are great too. There's a litany of them if you just search.
The "I need a ____" ones are especially popular. Just fill in blank with what you want to do.

I believe the GH series is a perfect first camera.

1. Teaches about different lenses via interchangeable lens system (though this is true of all dslr's) and adaptable M4/3 mount.
2. M4/3 size sensor grants more forgiving depth of field for handheld and beginners.
3. Customize-able options, accessories and (in the GH2, hacks) allow for a very long life in these cameras.
4. The adaptability of the lens mount means that you can look into purchasing less expensive lenses which still supply the glass quality you seek.

ONE THING I WILL SAY

I DO NOT think that a full-frame sensor camera is a good choice for a first camera for anyone who is considering working in the film industry. Though we may some day head that way, no serious cinema camera these days shoots on a full frame camera. Introducing someone to a full stills-35mm sensor is misleading and it will poison their perception with skewed views on "crop-factors" and low-light ability.
A beginner in the industry should be learning how to properly light a scene for the camera, not how to get a camera to cheat around poor lighting.

Start PAing.

Try to work with camera dept if possible. Eventually people will be working on their own projects and, if you've made a good impression, they will remember you for open positions.

Network, remember names and swap info/business cards.

Don't just network with people above you, other PA's as well, as they might reach higher positions before you do.

craigslist-for those with little to no experience and looking to get more
mandy-good for starting out
staffmeup-once you know what you're doing

Establishing shots when they present information or set up a scene. [A wide of a school before cutting to a classroom]
cutaways when it's in the middle of a scene [during an argument the character looks out the window, cutaway of what they see.]
inserts when it's a part of the scene but inserted in the middle of another shot [Someone is writing a paper and then there's an insert of macro pencil on paper]

The scene you describe sounds like a scene to me. It might be quick but if it's 4-5seconds long, it's probably there for a meaningful reason, in my opinion.

There are 3 routes I've seen so far.

1. Buy Camera, shoot things and then shoot bigger things...
2. AC. Start as PA, learn basics. 2nd AC, learn process. 1st AC, learn camera and lenses in and out, move up to DP.
3. Gaffer. This route is different because you need to learn the camera parts on your own but I personally enjoy working with these DP's best because they have the best knowledge of lighting.

mandy.com
craigslist is best for free