Daily Quest is a type of quest that appears during certain days only.
They can be useful when gathering strengthening materials for your Servants.
You can take Daily Quests by selecting "Chaldea Gate" on your terminal.
Senpai grew muscles all of a sudden...
Though growing muscles aren't really necessary, I recommend doing "Master Training" every Sunday.
Your maximum AP will increase each time you level up, which will allow you to take on more quests afterwards.
It might be a bit difficult to tackle "Master Training Advance" right away, but it's possible
to get some rare items from it, so you should try to challenge it once you've gained some levels.