For the most part, you'll need to keep on the move. It's too easy to get overwhelmed if you're standing still.
Make sure you've always got a card enabled. Your sword is okay against weak enemies, but once things start getting tough, you'll need some magic to get through.
You can double-jump and wall-jump by quickly tapping the jump button, and you'll need to evade the attentions of most of the bad guys.