If you're having trouble, here are some tips:
For horses, hippos, and ducks, just spam punch.
For hamsters, continually jab them from the front.
For giraffes and hydras, stand directly on top of them, repeatedly jump and jab them in the neck.
Don't just hold down the attack key; tap it repeatedly so you can do combos and move while attacking.
If you get knocked down, jump as soon as you recover, and always land within punching distance.
And kicking is absolutely worthless for everything. Never use it.
Adventurer: I'm preparing to slay a legendary monster!
Me: How about you go look for an old man's sack of clovers instead.
Adventurer: Okay.
Later...
Adventurer: Turns out he had the bag with him the whole time! Anyway, here's the quest loot.
*hands me a glowing sword and walks out*
Slaughtered my way down the dungeon, always being cautious. And then some kind of glitch happened which instantly drained my entire life gauge, despite no enemies being within attack range. No game-over screen. No titles, no badge. I am not very happy.