Now that the rain has (likely) cleared you can explore again. First, warp back to town and go through the far-left Lost Woods entrance. Why, it's a Mushroom! You'll trade this in a moment. Hit the pegs and go up to the left. You'll find a dwarf who wants to be taken back to his brother. First, warp to the witch's hut and give her the mushroom (she moves inside if it's raining). Equip it and press (not ) to give it to her. In return you get a bag of Magic Powder.
Now go a screen to the left and in the very far left corner is a bombable wall. Bomb open the rock, bomb open the next bit of rock (the peg heads and something else originally stopped you doing this) and get your Piece of Heart.
Warp back to the village and go into the forest and to the right. Go down into the stump for 40 rupees. Take note of that suspicious square of bushes above it—attack the middle bush and jump down the hole to get a Piece of Heart and hop down to collect those rupees and leave from the stump. You can return here at for a bomb-throwing Minigame with the friendly thief.
Go up to the right and down the stairs for 4 fairies and 100 rupees. Go to the very top right of the forest and go through the trunk to enter Dungeon 4.