You can choose to start your journey by playing one of four family members. Each person starts out with having different strengths but each character's stats will even out as they increase levels. The higher your level, the easier it is to catch fish. The highest you can reach is Level 99.
You will earn experience points as you catch fish. If you earn enough EXP then your overall level will increase. You will also raise the amount of Hit Points you have. Your HP value will determine how far you can swim. You can recharge your HP by sleeping at an Inn or eating some cooked food.
When naming your character you have to choose a First and Last name. The name you choose for your First name will actually be your last name and your Last name is your first name. The game was developed in Japan where your family-name comes first and then your own name, but that was not reversed when it was localized.
To save your game just hunt down the little save shrines. You will usually find them among the residential areas of the maps. You only have to stand in front of the shrine and press the X button to save your game. If you are playing as different family members you will have to make a separate save for each game you are playing.