This game utilizes Save Points in the form of lovely mystical circles which have no in-game explanation. Maybe Moogles did it, who knows?

Depending on their color these points do multiple things:
Blue save points are normal with no other options.
Green save points also restore HP.
White save points restore HP and MP.

It should be noted that if you come across a White save point, something dangerous is coming up. I for one hate fighting monsters only to get thrust into a boss battle with everyone wounded and drained so if you do enter an encounter like that you failed to save and should be ashamed.

You can also save anywhere in the Guild Hall or World Map.

Also, most non-dungeon locals allow you to "Dash" by pressing Shift. So if you're in a hurry or something you can move quicker.