Since missions can be long I'm wondering if I can save the game while "on the field". The first training mission says something about saving when you are sleeping on the bed but I wasn't able to figure out how so far.
I would assume not, since in every game in the past you couldn't. However, in your home village, you should have a bed which would allow you to sleep in. That bed, when slept in, will prompt you to save.
Hope this helps!
If you have a PSPGo, you can pause the game and perform a game state save. This is not the same as the game's saves - it's a feature of the PSPGo and can be used with any game, although the PSPGo only allows one state-save at any one time. Once saved, the PSPGo can be shut off completely without losing the state-save.
I believe the Vita also has this feature, although I have been unable to confirm this with the small amount of web searching I did.
Other PSPs can be put to sleep at any time. This has a similar effect except the PSP must remain with power - if the battery goes flat you will lose your place.