Adding to the above answer, if you were to shoot a bullet, you would also travel at the same speed and distance as the bullet you shot, due to Newton's 3rd Law of motion. Space, has tiny particles at different points, which will cause the bullet to slow down and stop but it would take a very long time.
I think that it would go the same speed as it is initially when shot on earth, but since there's no gravity or air resistance it wouldn't be slowed at all after being fired and would maintain that constant initial speed indefinitely.
But i guess there are other forces in space that would affect it like other planets....but I guess if we assumed it didn't encounter anything but empty space