Goron Bracelet
The Goron Bracelet allows you to pick up bomb flowers, useful in clearing paths as well as uncovering secrets. After entering Goron City, head straight down over the wooden platform in front of you and drop down to the bottom level. There is a door on your left with a rocky mat in front of it. Stand on the mat and pull out your ocarina. Play Zelda's Lullaby (on the yellow arrow keys it is: left, up, right, left, up, right) and the door will open. Run forward and speak to the Goron. After he finishes explaining how angry he is, bring out your ocarina again and play Saria's Song (on the yellow arrow keys it is: down, right, left, down, right, left). The Goron will start dancing and will reward you with the Goron Bracelet.
Piece of Heart
In the same room you obtained the Goron Bracelet, take out a Deku stick and light it using one of the torches. As quick as you can, run back out into the main area and light each torch surrounding the large Goron urn. The urn will start moving. Run up the stairs to the second level, just above the urn. On the left there will be another set of stairs where you will find a bomb flower. Pick it up and run back to the urn. When it is close enough, toss the bomb at the top of the urn. You need to time this right so that the urn is on the side closest to you when you run out with the bomb flower. Successfully land the bomb flower into the urn you will be rewarded with a piece of heart. Collecting four pieces of heart will give you one new heart container.
Opening Dodongo's Cavern
As you have been exploring Goron City, you will have heard of the problems the Gorons are having with monsters throughout Dodongo's Cavern. Dodongo's Cavern is the next dungeon in your quest and completing it will reward you with the spirit stone of fire. The cavern is located just below the entrance of Goron City and can be accessed after you have obtained the Goron Bracelet. A large boulder blocks Dodongo's Cavern, but by throwing a bomb flower down from above you will be able to get inside.
Big Bomb Bag
During the course of Dodongo's Cavern, you will receive a bomb bag, allowing you to carry a few bombs with you and making it easier to solve some of the puzzles. After defeating the cavern, you can obtain a bigger bomb bag in Goron City. From the entrance, head forward to the ledge and drop down one level. Watch for the large Goron rolling around the platform. Throw a bomb at the Goron, timed just right, so that it explodes in its path. The Goron will stop and offer you a bigger bomb bag. With it you can carry up to 30 bombs.