What would the horizon on an infinite flat surface look like?

If you were looking straight ahead, so that your line of sight were parallel to the ground, then the horizon line would appear to be at eye height, in all directions.

Upon your appearance (by whatever means) in that place, you would instantaneously be flattened (to a thickness so infinitesimal that it would be inherently imperceptible), by the gravity of whatever substance or energy comprised the infinitely wide and infinitely long (but totally flat) surface, unless your mass would (somehow) be of an equal or greater magnitude: In that last-mentioned case, you would be integrated into that surface, or vice-versa. (The lack of any relation between the rate of these changes and our own speed of light--in vacuum or otherwise--would be required by the fact, made very plain by Einstein in his 1916 pop.-sci. version of General Relativity, that the speed of light is local: The locality you've described could not exist anywhere, as, if it did, it would literally exist everywhere, per your definition of it as infinite.)

Any flat-earth survivalists among interstellar tourists will note that the horizon of planets much larger than earth is a much longer stroll from them than earth's horizon had been when they boarded the rocket. But it will still be infinitely closer than an infinitely flat universe's horizon could (even in principle) be.