In short, there's two aspects to super training: the minigames and the punching bags. The minigames are the main part of super training and come in 3 levels plus a secret level (unlocked after you max out a Pokemon's EVs, those stages give out rare items instead of EVs). I suggest taking some random Pokemon you don't care about and clear ALL the minigames for at least the first two levels so you have them all unlocked (to unlock level 2 you have to clear all level 1 levels, same for level 3 but with 2. Clearing all level 3 games doesn't unlock anything).
When you play a minigame, if you win you get a certain amount of EVs in the appropriate stat (4 for level 1, 8 for 2, 12 for 3) as well as a punching bag, typically of the same stat with better performance in the minigame giving a better bag. EVs also determine how well you perform in the minigames, the ones I know are that special attack and attack increase the damage your hits do while speed increases the speed of the platform you move on. I believe the defenses improve the charge rate of your shots while HP decreases the points penalty from being hit. Don't worry about those though, just focus on whatever EVs you want to train. I do suggest starting at level 1 then moving on to 2 and finally 3 even when you have them all unlocked though: the level 3s without any EVs at all are pretty tough.
Now, the punching bags are pretty much just as important as the minigames though. You can hold up to 12 of them (plus whatever one you're using at the time) and they're single use. While your Pokemon will very slowly punch it on its own (like once per minute) it's a much better idea to tap on the screen yourself to speed it up. The "blank" punching bag gives you a random bag occasionally after enough hits or you can get bags through the minigames as I said above. There's bags that give you EVs (1 for S bags, 4 for M, and I think 12 for L), ones that give you a boost in the next minigame you play, and perhaps most importantly the reset bag. You can get a reset bag by hitting the blank punching bag until the RNG blesses you, and you can use it to reset all EVs to 0.
Yes you can still gain EVs the old way, however Super Training is much faster and arguably more entertaining since you don't have to worry about what Pokemon you fight. It's also much faster and you can do it right from the start, I fully EV trained my Fennekin in about an hour from scratch. It can start off slow but gets really fast as you gain more EVs and the minigames become easier and easier.
Oh, also, EVs max out at 252 this time, no need to keep track of them at all and you can see your EVs in a chart on the super training page as well as how many EVs your Pokemon has total in the green bar.