"Toward" or "towards" – what would a native speaker use?

Right - American English favors "toward," and UK standard tends to add an 's.'

This also applies to "forward," "backward," etc. These are explained in more detail by Fowler, Merriam-Websters Dictionary of English Usage, et al.

Let's quote here the “Word note” about this issue in the New Oxford American Dictionary:

toward, towards: It might seem pedantic to point out that toward is the correct U.S. spelling and towards is British. On the other hand, so many writers at all levels seem ignorant of the difference that always using toward is a costless, unpretentious way to signal your fluency in American English. It's the same with gray (U.S.) and grey (Brit.), though many Americans have been using these two interchangeably for so long that some U.S. dictionaries now list grey as a passable variant. This is not likely to happen with toward/towards, though—at least not in our lifetimes.

_{It is followed by a note: “Conversational, opinionated, and idiomatic, these Word Notes are an opportunity to see a working writer's perspective on a particular word or usage.”}

As a native speaker of british english, to me, 'towards' implies motion - so I would probably use them like this:

"He faced toward the house"
"He walked towards the house"

only adding the 's' when associated with a verb that includes movement. I don't know if that's correct in any official sense, but that is how I would use them in ordinary speech.

It seems that since the two are interchangeable you can pick one you like better and use it. I doubt a native speaker will even notice. Try to be consistent, I guess. Or, if you'd rather fit in as much as possible, try to listen to the people around you and imitate them. But for this one word you might have a hard time finding out if you're with a bunch of '-s' using people or not.