Each character can give multiple rewards through bond level, which are mostly Spirit Crystals and Home screen messages. You get bond experience by using the character for quests.
To view and claim bond rewards, go to the "Character Bonds" option that's available in the menu. Tap the "Collect All" button on the bottom right of the character bonds page to quickly get all the rewards from all characters.
Reaching max bond level 20 gives a total of 650 or 700 Spirit Crystals depending on the Ally.
The "New Home screen message" reward is another dialogue that you can check when you talk to them on the home screen. The Allies that shows up on the home screen are random, but you can keep refreshing it by going to the menu and selecting "Home" again.