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To get all pokemon people usually catch and evolve all they can in their game then trade over wi-fi for the rest. Breeding for stats is more for competitive play to get perfect IVs and egg moves. If you want to know what IVs, EVs or natures are just check on bulbapedia or serebii or something. Simply put though natures give a 10% boost to one stat and a 10% reduction to another (mostly), EVs make it so depending on what pokemon you train against, you gain different stats (killing something like Audino over and over gives you HP EVs so you will have higher HP than you would otherwise), and IVs are your natural potential in each stat, randomised from 0-31 when the pokemon is made and can not be changed. Higher numbers will make them better, for example a 31 speed IV will mean it can be faster than any other pokemon of the same type (excluding EVs and natures which may also effect it).
For normal in game play I usually just make sure the natures of my team are good and then keep EVs in the back of my mind so if I'm fighting a lot of things that give attack EVs I'll try to use something that needs high attack. I don't worry about IVs.
Also play pokemon showdown for competitive battles, it's all the fun with none of the hastle.
