I'll let someone else handle the mathematical approach and just describe the general results.
I'm in orbit and I fire a bullet...
I'm going to assume you're in a circular low Earth orbit at 200km altitude to start. In each case, the bullet fired continues in an elliptical orbit that intersects the point at which it was fired.
(1) Along the direction of the orbit forwards (maybe achieving escape velocity)
The bullet enters an elliptical orbit, with perigee at the altitude of your initial orbit and a higher apogee dependent on the muzzle velocity of the bullet. A pistol bullet (muzzle velocity around 360 m/s) would get to about 1600 km; a rifle bullet might reach a 4500 km altitude. The fastest bullets fired from hand weapons have a muzzle velocity around 1200-1400 m/s, which is not sufficient to escape Earth orbit, or even reach the moon.
(2) Along the direction of the orbit backwards (maybe cancelling out forward velocity entirely causing the bullet to drop vertically)
Muzzle velocity a fraction of orbital velocity (~7800 m/s), so the bullet won't come to a dead stop, but it will slow enough to drop perigee to below zero altitude, so the bullet will re-enter the atmosphere and burn up in less than half an orbit.
(3) Directly toward the planet
(4) Directly away from the planet
In either case the bullet goes into an elliptical orbit with a lower perigee and higher apogee than the original. I don't know off the top of my head how to calculate the altitudes, but the perigee is probably in the atmosphere, so the orbit will rapidly decay.
(5) 'Sideways' in the direction of the poles
This changes the inclination of the orbit by a degree or so while leaving it nearly circular.