So every time your pokemon "kills" another pokemon gets points called EV(for example defating Bulbasaur gives you +1 point to Sp. attack EV). You can accumulate up to 510 points total, but only up to 252 in single stat. Due to way this EV are "translated" to stats only number dividable by 4 makes difference(Which means there is no difference between for example 1 point and 3 points).You speed up EV training by equiping bracers(For example if you are holding power anklet you are getting +8 speed and +1 Sp.Attack for killing the same bulbasaur) at cost of being 50% slower. You can buy them at 4th floor of Departament Store. You can use "Friendship berries"(Avaiable at second floor of departament) to remove up to 10 point from single stat per use making it available to be allocated elsewhere.
So let's look at your questions.
"fighting certain pokemon give certain EV's but how should one go about this? Do I have to perfectly construct my team by only making them fight the specific ones that raise that stat?"
You can try to optimize process of EV grinding by fighting only specific pokemons,but way I approach this is I grind as normal with bracers equiped(then I got more points in stat I want than in unwanted stats). And then use berries to remove EV I got in unwanted stats. I recommend Obsidia Slums for EV training since pokemon there gives only HP,Attack and speed EV(meaning extra points for stat you probably want and not many stat to clean with berries).
"Is there a specific level I should start ev tranining at?"
The faster you start the better,since you will get less useless EV and spend less for berries. I recommend ev training when you get access to bracers.
