Daily Quest
Preface
Each weekday consists of same-attributed mobs for the simulated battles, which yields stars (corresponding to the day's attribute) to level up companions. On the flipside, each weekday's skill training consists of randomly-attributed mobs, but all are users of that day's weapon; these quests yield items to increase a companion's skill level.
However, on weekends, 
ALL-ROUND stars drop in the simulated battles, and coin battles replace skill training quests which yields large amounts of coins instead.
Keep time zones in mind, as Japan tends to be a day ahead of the West.
Monday
On Mondays, you can collect 
AETHER stars and increase skill levels of 
Slash users. In ordinance with the attribute advantage chart, while caution is required, 
NETHER companions are best to tackle through this day's simulation battles, and 
Shot or 
Magic users for the skill training to avoid being slashed by attacking from a distance. Monday is also the day to collect materials to limit break AETHER companions.
Tuesday
On Tuesdays, you can collect 
FIRE stars and increase skill levels of 
Thrust users. In ordinance with the attribute advantage chart, 
WATER companions are best to tackle through this day's simulation battles, and Slash or Magic users for the skill training to avoid being pierced by attacking from the sides. Tuesday is also the day to collect materials to limit break FIRE companions.
Wednesday
On Wednesdays, you can collect WATER stars and increase skill levels of 
Blow users. In ordinance with the attribute advantage chart, 
WOOD companions are best to tackle through this day's simulation battles, and Slash, Thrust, Shot or Magic users for the skill training to avoid being striked by attacking from the sides or a distance. Wednesday is also the day to collect materials to limit break WATER companions.
Thursday
On Thursdays, you can collect WOOD stars and increase skill levels of Shot users. In ordinance with the attribute advantage chart, FIRE companions are best to tackle through this day's simulation battles, and Slash or Magic users for the skill training by attacking from the sides. Thursday is also the day to collect materials to limit break WOOD companions.
Friday
On Fridays, you can collect NETHER stars and increase skill levels of Magic users. In ordinance with the attribute advantage chart, AETHER companions are best to tackle through this day's simulation battles, and Slash users for the skill training by attacking from the sides. Friday is also the day to collect materials to limit break NETHER companions.