Your food resets every time you go into a dungeon. You gain more food my flipping food tiles which share the same color as the relationship tiles, you can also buy rations in the shop which gives you 5 food. Each turn per unit uses 1 food (you can see this in the upper right corner when in a dungeon). Various decisions in the game affect your relationship percentage with each girl so pay attention to the dialog if you are really worried about it. You can save in the city by pressing "V" and going into the phone, there is a save button and a save and quit button at the bottom. The phone can also be used in dungeons to use items in between fights. The game autosaves before every tile in dungeons so you can easily cheese rng wheels by restarting the game and loading the autosave, if you dont get the outcome wanted.
EDIT: P.S you can buy thread at the shop for pretty cheap that allows you to go back to town from a dungeon without having to move all your units to the yellow tile. This is helpful for completing missions to get the three gold tickets and it is also helpful for maintaining relationship percentage because if you fail a dungeon because you ran out of food. If you fail a dungeon due to food you will take a hit to your relationship to your equipped units. You can use the thread by using it in your inventory whilst in a dungeon.