There is no trick or fast way of reaching the maximum relationship level. This will unlock for you as you keep progressing the story and doing as many side objectives as possible.
Each piece you gather for the ship should give you level up.
Each significant battle will also reward you with large amounts of relationship XP. This can range from optional bosses (3 of them) or large monsters on an islet that count toward cleaning an islet.
Each optional conversation and actions in the base (talking on the couch, eating several meals and then sleeping) will also award XP. Make sure you talk to Yu & Kay inside the base as often as possible and also interact with the collectibles you find come across in abandoned buildings/houses.
Camping and eating can be "abused" in such a way that if you eat food that doesn't restore a lot of hunger, you can camp/eat several times to trigger more cutscenes if you're missing any. I wouldn't really recommend this until you've unlocked all the fruit and have the ability to make every type of dish in base and by the campfire. Each dish will provide a new conversation the first time around and any repeat dishes should help push optional scenes along (playing the guitar, turning off the spud toy.. etc..).