To get some characters’s weapons, costumes, and accessories in their stages, you will need the assistance from a second player (on Xbox Live) using a different character.
When you encounter character specific obstacles in a character’s stage, you will need the special abilities of a character to get past them.
In short:
Galen is used to destroy the pillars/rocks with the orange lightning bolt/crack. When you see obstacles with this symbol in a non-Galen stage, put Galen as a second player so he can remove those obstacles for you.
Maggni is required to move the white-stone pillars. When you see one of these in a non-Maggni stage, put Maggni as a second player to move these rocks into place.
Sephia can douse orange and blue walls of fire. When you encounter these immobile curtains of flame in a non-Sephia stage, put Sephia as the second player to douse the flames.
Levv can latch onto the pulsing purple orbs with his grappler. There seems to be almost no stages that require Levv’s special ability.
Zazi can shoot (press Block or Skill Use Toggle in first-person view) the crystal switches for gates and bridges. When you find these crystals in a non-Zazi stage, be sure to put Zazi as a second player to activate them.