You need to be able to shoot arrows with the bow to enter the second dungeon. You need to shoot an arrow at the blue shield like object which it looks like you did. To enter the water parts, there go into the forest north of the starting village and then go west. You need to burn some bushes to get through (stuff like this comes from the original NES Zelda). You need to upgrade your purse and you can't upgrade it until you beat the 4th dungeon.
If you want a video walkthrough, go to Zelda Classic → Isle of Rebirth Walkthrough: 1 - Level 1, Seaside Ruins - YouTube. It might be a lot better than me explaining things to you all the time. They didn't play hero mode though.