First, the barrel does not "climb" for any reason inherent in the barrel. Any firearm recoils around its own center of gravity (or the center of gravity of the gun and the hands or arms holding it). Since most guns have the barrel above the center of gravity, the gun muzzle will recoil upward. Guns have been made with the barrel below the CG; the barrel "climbs" downward.
Recoil begins the instant the bullet starts to move. Since the gun weighs more than the bullet, the bullet may exit before there is much gun movement, particularly in a handgun, where the weight difference is less.
To see how much the barrel will move during recoil with the load for which the sights are set, simply lay a ruler along the sights, and notice that the line of the sights is at an angle to the line of the barrel. That difference allows for recoil.
Jim