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I have never done this, so I am not sure where to begin.....
I think this is called laminating, which covers both sides of the paper.
I'd be happy to just cover the front side of the page too, unless I
double side the printing, (which I probably wont do, because I am not
sure how to do it on a home printer).

Anyone know anything about this plastic coating?

Con-Tact (the makers of "Contact paper") make a version which is a
thin clear vinyl, with adhesive on the back. It's thinner and more
flexible than the usual "lamination" plastic, and it doesn't require
heat to apply (as heavier lamination materials do).

The one I'm looking at here on-line on the Staples web site is thieir
item 401015, model C9AC12-06... a roll 18" wide, 20' long, for $9.
It's described as "repositionable" but I would consider it it
permanent when applied to something as thin as paper.

I've used it for years as a way of putting a protective layer on
pieces of paper that I wanted to keep in my wallet. It has a matte
finish, and is thin enough that it doesn't obscure laser-printed or
hand-written text on the paper.