How do I save my progress in Dragon Age: Inquisition?

I can't say the for sure on the xbox, but for PC you can't save until you have completed the Character and gotten through the first cutscene in prison.  Once there, you access the character menu and choose the option save game.  It also auto saves at different points, fairly often in fact, while playing the game.

If the option isn't there to save in menu...do you have enough free space to save??  Other than that, repair option or reinstall??  You may also try taking the game offline and see if it effects anything(by offline, I mean disconnecting completely the internet).  The keep isn't for keeping saves, it is for tracking gameplay for future games time continuation, and to allow you to have custom world states from previous games decisions.  There is supposed to be a cloud save ability, but I don't use it on my PC, and some poeple say it is good to use, other say it causes problem.