On a general level, if you want to produce a "Pokémon Breeding Process", it ought to actually result in something special. Having 31 IVs in all six stats is special. Having them in one, not so much. Somehow raising the chances of finding a Shiny? That's special. Raising the chances of getting a "crap" Pokémon? Not really.
To consider what Pokémon breeding is, we must consider how animals reproduce. On the superficial level, it generally requires a male and a female: such holds true here for the most part. Unlike normal, though, Pokémon typically lay Eggs to bear baby Pokémon.
Normally, Pokémon breeding is more than simply about getting lower-evolution Pokémon or spawning massive numbers of starters: in fact, it is the most certain way to guarantee that your Pokémon can get the most ideal stats and movesets possible!
However, this particular primer is designed for Pokémon beginners - if you want to get into the gnitty-gritty of things, the other sections will sate you.
As you progress through the game, you'll eventually hit Route 7. After some events there and at the Parfum Palace, you'll be allowed to reach the majority of the route past that big bridge. Just beyond said bridge is the Pokémon Daycare. There, you can leave two Pokémon and they'll stay there.
Typically, beginners just leave Pokémon there for the purposes of leveling up: they'll earn 1 EXP. per step you take. You can get the Pokémon back at a base cost $100, plus an additional $100 per level earned. You, however, cannot dictate how moves are learned or deleted, and level-up evolutions do not occur, so look out!
Now, Pokémon breeding... Obviously, you'll need two Pokémon: one a male, and one a female. (The little mark that appears beside their name in battle represents that: the blue arrow is male, the pink cross is female. No mark means the Pokémon has no gender and generally cannot breed.) There are blatant exceptions to this rule, but more on that momentarily.
When you leave two Pokémon there of opposite genders (or other situations), there is a chance that you can speak with the man outside the Daycare to receive an Egg.
The Pokémon within the Egg can inherit a number of things, depending upon certain conditions - these include IVs (which influence stat growth), Nature (which can give a 10% bonus/loss to some stats), moves (but only some!), and species.
So, we come down to it - how to make the Egg hatch? The Egg hatches by carrying it in your party and walking/biking/skating around. Eggs can hatch in anywhere from 5,000 to 31,000 steps, depending on the species within the Egg. It's just walking, so beginners don't really need to worry.
Eventually, the Egg will hatch. This process can be sped up by having Pokémon with the Magma Armor or Flame Body abilities. That's the gist of it. But if you want to manipulate this in a more beneficial way, read on...