I thought I'd do a blog on how to get Pokémon with high stats. Different portions of it apply to different Games, so make sure to check what Games each of these applies to.

1. Get the Right Nature
If a Pokémon has a Nature that raises a Stat, it's Stat will ALWAYS be 10% higher than an identical one that doesn't. The same is true of the Stat each Nature lowers. It's always 10% lower. Even in Gen 7, there is nothing you can do about this. As a result, Nature is very important.

The best way of getting the Nature you want in a Wild Pokémon (including Legendary Ones) is to have a Pokémon with the Ability Synchronize at the front of your Party. For anyone interested in having good stats, the first thing you should do is catch a bunch of Pokémon with Synchronize (like Abra, Ralts, or Munna) that have Natures matching the ones you would want.

For Breeding Pokémon with the right Nature, you need one of the Parents to have the right Nature. Give that Parent an Everstone before breeding.
(This works perfectly in the newer Games, but only raises the chance in Gen 4 Games, and in Black/White.)

2. IVs (Gens 4-6)
For a person concerned about their Pokémon's Stats, these are arguably the most important (and annoying) part of it. If you play in a Gen 7 Game, thankfully, you don't have to bother with this (provided you Level your Pokémon to 100), because of Hyper Training.

For everyone else, however, these are a huge hassle. Basically, when you catch a Pokémon, each of its Stats vary within a range, which is based on the Pokémon it is. Where they are within that range is referred to as IVs.

Unfortunately, you can't change them in older Games, nor is there any good way to get Pokémon with high IVs to appear in the Wild (aside from the DexNav in ΩR/αS). It's basically a matter of repeatedly catching the same Pokémon, hoping to get a good one.

For those who don't want to do this, there is a "technically" legit trick, called RNG Manipulation. I won't get into it, other than to say it's only legit if you do it WITHOUT MODIFYING YOUR SYSTEM OR GAME CARD. (Which is only possible in Gens 3 & 4, possibly 5.)

Also, in Gen 6 and beyond, ALL Legendary/Mythical Pokémon will have at least 3 of their 6 IVs maxed out, which is nice.

For non-Legendary Pokémon, Breeding is the best way to get the IVs you want. I'll probably have to make a blog just about Breeding, but for now I'll just say that you can use certain Items (the "Power Items", like the Power Anklet, as well as the Destiny Knot) to pass down Stats from Parents.

3. EVs
These are what items like Protein and Calcium raise. Every Pokémon can earn up 510 EVs, through battling other Pokémon. Up to 252 (or 255 in older Games) can be earned in each stat. Once you reach 510 total, you can't earn any more. I will give advice on how to raise (and lower) your EVs in each Game.

In Gen 4, the best way to earn them is to find some common, low level Pokémon, that raises the stat you want, and repeatedly defeat it. Using Power Items speeds this up a lot, as does the Pokérus.

For lowering EVs, just grow a bunch of those Berries that lower a Stat and raise Friendship.

In Black/White it's pretty much the same, except it's harder to get Berries.

In B2/W2, however, it's a lot easier to lower EVs. In Join Avenue, there are Shops that sell those Berries (at high Ranks), and also Salons that can lower EVs.

In Gen 6, not only can you SEE your EVs (in the Super Training screen), but you can also raise them A TON faster. Super Training obviously works well (and it's REALLY FUN), but there are even faster ways. The fastest way to earn EVs is to find a Horde of Pokémon that raise the Stat you want, and take them all out in one hit (with something like Surf). This Raises your EVs 5× faster than taking out one Pokémon at a time. Additionally, the Exp. Share works differently now (similar to Exp. All from Gen 1), and you can work on your whole Party's EVs at once! Add to this the fact that they can now hold a Power Item AND use the Exp. Share, and you can raise EVs EXTREMELY FAST! (And don't forget the Pokérus, too.)

That technique works in X/Y, and ΩR/αS. However, there's another good option in ΩR/αS. You can invite certain kinds of Trainers to your Secret Base. Then, they can "Exercise" your Pokémon, raising EVs. It's not the fastest way, but it doesn't give Exp, so it's good for raising EVs without Leveling your Pokémon.

In Gen 6, You can grow Berries, but there's also another option for lowering EVs: The Reset Bag in Super Training. It's also good for measuring your EVs. (DON'T FORGET TO SAVE FIRST!)

In Gen 7, Isle Evelup is the best way to raise EVs. If you leave Pokémon there to Train (and use Poké Beans), you can max a Stat on 18 Pokémon in less than 16 hours (at Rank 3).

Some Food Stalls in Festival Plaza can lower EVs, or you can grow and use Berries.

You can check your EVs by pressing Y on your Stats Page.