Mounting and aiming the scope are covered in the instructional videos linked, I'm sure.
What is the purpose of zeroing your rifle: hunting, target competition at set distances, just so you can hit targets for fun at the range?
You don't shoot a rifle flat (initial trajectory parallel to the ground), because the projectile will fall off quickly and you won't get very much range. Instead the barrel is pointed up a very slight degree, which makes for a rising trajectory, an apex where it reaches the highest point, then a constant dropping acceleration due to gravity (the velocity is also constantly decreasing due to drag).
Thus a zero for 100m might show -4 (below the bull) at 25m, -1 at 50m, +1 (above) at 75m, and 0 at 100m. Note that this example also has a second zero on the rising leg of the parabola around 60m.
Problem is, every cartridge is different! You need ballistic data for the load (velocity/pressure) and projectile (aerodynamics), which can be determined experimentally through many, many shots and meticulous date recording and calculation. Thankfully, ammunition manufacturers might include ballistic tables, either on the box or through their web site. There might be a recommended zero with a chart of distances and impacts above and below at those ranges, or else the velocity, bullet weight, and coefficient of drag, which must be plugged in to trajectory formulæ (or a ballistics application) to plot the flight.
Once you have the data, it is possible to zero a scope for a completely different distance than that you're shooting at. Many people will zero at 50m rather than 100m or more, because the holes in the paper are easier to see, as well as a much reduced effect of wind at lesser distances. For fun shooting you just want to get it on target and adjust if needed. For competition you'll want to get it dead on zero for a known distance, possibly with known adjustments if shooting at additional distances.
For hunting you might zero at +2 at 100m instead of dead on, which puts a zero on the near side at say 55m and one further at say 150m. This assumes the intended target zone might be a four inch circle on a deer, so effectively any well-aimed shot is within that 4" (±2") at any range from spitting distance out to 150m, and beyond to 200m where the drop is -2". This is the concept of point-blank range: the range at which sight adjustment is not necessary.
When you think you are pretty close, shoot a 5 round group. Then make the final adjustment. Trying to zero while adjusting after each round is an exercise in futility.
Yes, you zero groups, not shots. Unless your rifle is set up very near perfect, and your form is exactly perfect, no gun consistently hits the same hole every time; there is a natural average spread. (Now this spread can be small, even just 1 MoA in many modern off-the-shelf hunting rifles.) If your first shot is to the far left of the natural spread, when you adjust the scope to what you think is an anomalous shot you've actually shifted your average centre point to the right. Now the next shot could be to the far right of the natural spread, and it looks like you're twice as far off in the other direction now! Don't chase adjustments on individual shots.