The anime cliche I hate most? "The heartless killers defeated me, but I was just knocked out,because I'm one of the good guys. Now at just the right moment, I'm back." You don't see it that often but you see it often enough to get on my nerves. They did it a lot on gunxsword. A few times on Bleach and One Piece. Annoys the heck out of me.
As for training, all those Animes remind me of the Rocky series of films. Compare DBZ or Bleach to Rocky. Rocky is living a normal life. Circumstances force Rocky to face a powerful foe, luckily for rocky there is something special about him (his body blows can break a cow's ribs). Rocky finds an old pro to train him, and rocky also meets a girl. Because a girl is always needed for our hero to fight for and because it's great drama when we can see her suffer at his misery.
After some more training Rocky eventually beats the first villain, through a mix of new abilities and sheer force of will. This first big villain becomes Rocky's friend, helps him train and also fights by his side (Renji, Piccolo, Vegeeta ect). Rocky fights some other colorful villains like Thunderlips and MR T (who gets a training session after he defeats rocky, who then defeats him) then he meets a new powerful, murderous foreign villain who defeats his best friend and also wants to rule the world. Rocky knows what he must do. Train harder than ever. For a whole three montages rocky trains to defeat Draco. When it is over, he's outrunning cars and shit. He has clearly reached a new level of power. And by the way, we can actually see a fighter's power level in this movie, because they punch this pad that shows us how strong they are. This pad would later show up in DBZ. and while I know fighters must punch these pads form time to time I can't help but feel it showed up in DBZ because Toriyama was a fan of Rocky and saw it there.