First of all, IVs are a hidden value for each stat in a Pokemon, namely: HP, Attack, Defense, Special Attack, Special Defense, and Speed. This hidden value can range from 0-31. It will be added to the Pokemon's respective stat when it gets to Lv. 100. For example a Shaymin with 0 IVs in Special Attack compared to a Shaymin with 31IVs in Special Attack. Ignoring all other factors such as natures and EVs, if 0IV Shaymin has 250 in Special Attack, 31IV Shaymin will have 281 in Special Attack. The computation for HP is quite different though if I remember correctly. A Pokemon with 31IVs in every stat is called a Flawless Pokemon.
Now, since I mentioned EVs, these are points received depending on which Pokemon you defeated. For example, a Chespin yields 1 Defense point when defeated. For each Chespin you defeat, you earn 1 Defense Point and for every 4 Defense Point, it is +1 to the Defense stat when it is at Lv. 100. However, for each stat, you can only earn a maximum of 252 EVs Divided by 4 is 63, so if you EV train a Pokemon in Defense to the max, you will have 63 more Defense at Lv. 100. The total maximum EVs for a Pokemon is 510, only enough to max out 2 stats and add one to 1 stat. This can however be mixed across all stats, but it isn't recommended. Ev Training can be made easier with Power Items, Macho Brace, and Pokerus. Macho Brace doubles the points earned just like Pokerus. Pokerus+Macho Brace will make it X4 Power Items on the other hand add 4 EVs to their respective stat on top of the EVs earned by defeating a Pokemon. EXP Share can also be utilised when EV Training. All Pokemon in the party will receive the EVs.
Natures, each nature raises one stat and lowers another. This is with the exception of Docile, Bashful, Serious, and Hardy which don't raise or lower any stat. These are called neutral natures and are not usually used competitively. For the usual natures, the stat increased is multiplied by 1.1 and the lowered stat is multiplied by 0.9. Adamant Nature for example multiplies the Attack Stat by 1.1 and Multiplies Special Attack by 0.9 the boost and reduction can be seen when looking at stats in summary, the increased stat is in red text while the reduced is in blue text. Why want a stat lowered? Well it is because some stats are not really used in battle, like for a physical attacker Blaziken, you won't be using SP Attack so might as well sacrifice that for Attack or Speed. For mixed attackers, you may need to sacrifice a defense stat(Defense or Special Defense) or speed.
For breeding, you can always have the same nature. This can be achieved by giving an everstone to the parent with the desired nature. Regularly, breeding passes down 3 random IVs from both parents, but if you let one of the parents hold a Destiny Knot, the offspring will have 5 Random IVs from both parents so if you breed 2 flawless together and one of them is holding Destiny Knot, all eggs will have at least 5 Perfect IVs. Hidden abilities can also be passed down if a male, bred with a Ditto, has it or a female, bred with a male of the same egg group has it. Balls are always inherited from females. When bred with a Ditto, offsprings will always be in a Pokeball. Egg moves are moves that can only be learned by breeding. For example, breeding a Male Clawitzer with Aura Sphere with a female Squirtle will yield a Squirtle with Aura Sphere. Offsprings will always be the same Pokemon as the female. When male breeds with Ditto, it will always be the same as the male.
Good Luck with Breeding and Trading
