Your hit points are directly related to your fishing skill; the more hit points you have, the further you can cast your line. Hit points are also needed to keep you from drowning when you're diving for shellfish. Each brother will start the game with a different amount of hit points. The Kid will begin with 50 hit points and the Teen will begin with 80 hit points.
To gain more hit points you will need to increase your level. You can do that by fighting monsters that will randomly appear and attack you. That isn't the only way to gain more hit points though. There is a lady who will give you 10 hit points by bringing her 5 Green Grasshoppers. The other one is to complete your mom's flower bed by collecting all 32 flowers and then talking to the girl by your house, who will give you 40 hit points!
When you run into a monster the battle screen will appear. The three options you have are BEAT, EAT or RUN. Selecting BEAT will bring a pointing hand on the screen the hand will move around randomly and when it is positioned over the monster press the A button on your Gameboy. Depending where you hit the monster and how much hit points you have, you'll damage the monster. EAT will take you to your food menu where you can eat as much as you need to regain hit points. RUN makes you run away, but sometimes you can't run.
Each map area has different types of monsters. The farther out you go, the harder the monsters get. The enemies in the Summit have a lot more hit points then the ones in the Stream.