We do send them both left and right (west and east), as well as north and south. That’s how you get to orbit. Mainly east to take advantage of the earth’s rotation (can’t think of a good reason to send them west, other than suborbital/ballistic flights. Maybe it’s been done for orbitalrockets, I never checked). North or south for polar orbits.

Each of those flights start by going up in order to clear the thickest part of the atmosphere. If you watch a rocket launch long enough, you’ll see it slowly start pitching down after a while in order to gradually oriented towards horizontal flight, this is called a gravity turn.

As for down, well… you will not go to space today.