Instructions
Start a game, then complete the initial training missions. At the end of these missions, you'll be given a headquarters called "Mother Base." On this screen, tap the "R" button. The first option here is the "Save" option.
Complete the mission you're currently playing. Press "X" at the "Results" screen, which will take you back to Mother Base. This auto-saves the game.
Lightly tap the power button to the right, rather than holding it. This will quick-save the current mission. Be warned that this lasts only as long as the battery holds out in sleep mode, so take care to finish the mission in order to save properly.