If you fly a kite where the wind is horizontal, how does the kite know to go up?

google "kite aerodynamics"

It is called lift. The same reason that an airplane goes up when you increase power and angle of attack. The kite is set up to do that.

Hold the kite with the front end up (positive angle of attack) and lift is generated and together with the force of the string the kite goes up.

Angle of attack, same as a wing.

It's pretty much the same quandry as how a symmetrical wing could possibly lift an airplane. Give either one of them the right inclination, and up they go.

So if I hold the front end to my left (left angle of attack) will the kite fly to my left, or will the kite fly up? If the kite will fly up, how does it know which way is up?

Which way is up?

It neither knows, nor cares. Neither does a wing know or care. But the pilot does. And both a wing and a kite are directed in the direction the pilot wishes. Neither the kite nor the wing have a vote.

Simple kites don't have steering. Kites that steer respond to however that steering is designed. The only one I've seen that steers would steer to it's right, the pilot's left, if the left strings were pulled. It did so because it's lift vector was yawed from straight away from the pilot when the kite was yawed.

It doesn't. But gravity does. If the kite has the proper Center of Gravity to Center of Pressure relationship and if the kite is bridled correctly, it will stay pointed up. If stability is marginal, a tail will also help.