Story mode is episodic, with each Episode being composed of a single battle wedged between some dialogue sequences. You will have Ion's first Episode unlocked by default. After beating the first episode, in which you fight Kirio, the first of Kirio’s Episodes will become available. This is the general pattern for unlocking episodes of other characters. When Ion fights a “new” character, the first of that character’s Episodes become available. For this reason, I’d personally recommend beating all of Ion’s Episodes before moving onto another character, so that you have all characters available in Story mode.
In certain cases, a character will have additional Episodes which cannot be accessed and appear to be locked, even though you’ve beaten the Episode previous to it. This can be seen in Kyouichi’s Episode sequence. His third Episode doesn’t open up until you’ve gotten to a certain point in Raika’s Episode sequence, due to plot progression.

While you may have a character’s Episodes unlocked in Story mode, this doesn’t mean that you’ve unlocked them. By default, you have Ion, Kirio, Aoi, Leafa, Raika, Kyouichi, Iks=Piari, and Lazona available for use in other game-modes outside of Story. There are a total of six characters to unlock, four of which can be unlocked through the “usual” method. This simply refers to beating all of the applicable character’s Episodes. For the sake of convenience, I’ll just list the Final Episode of each of the four characters that can be unlocked in this manner:
Mersa Gloria: ADHESION
Claudio Vecela: To Tomorrow’s Skies
Avain Clane: Father and Daughter
Albis: The Promise
Upon finishing one of these episodes, you’ll get two Achievements simultaneously; one for clearing the Final Episode, and one for unlocking the character. (Typically, you’ll get an Achievement for finishing a Final Episode belonging to any character.)
As of the time this guide was written, I’m actually unsure of how to unlock the final two characters (Corrupt Ion and Overdrive Claudio). Allegedly, they are unlocked by completing the Magical Battle Festa mode without being defeated. It’s unclear if they are unlockable if the “Restart” option is used. Since I haven’t unlocked them (yet), I cannot vouch for the validity of this method.

This will likely be the shortest section of the guide. After clearing all of Ion’s Episodes, you will unlock two modes: Survival and Magical Battle Festa. Survival is just what it sounds like; trying to make it through as many battles as possible without being defeated. Magical Battle Festa mode is a tournament-style mode against computer-controlled opponents.
If there is another mode (or modes) that I am unaware of, I apologize for skipping it.

Since the method of unlocking Formations and skills is almost identical, I’ll just put them in one category, which should also be very brief. After clearing a character’s Final Episode, various Formations and/or skills will be unlocked. I don’t believe that the Formations/skills are character-locked, but rather unlock in a predetermined order. After you’ve unlocked all of the Formations, you will be given an Achievement and the game will switch over to unlocking only skills after you’ve completed a Final Episode (meaning that it unlocks many more skills than it did when you were still unlocking formations).