Getting across the broken bridge
You can get Epona and ride to Gerudo valley, get her up to full speed, and jump across the bridge. However, it is much easier if you have the Longshot. Use the targets strung on rope over the bridge ends. Once across, talk to the master carpenter outside the little white tent. He will tell you that the bridge was destroyed by Gerudos. Inside the tent, you will find the Hyrule Field runner. He won't do much for you now, but once you finish later sections, he will present a rather difficult opportunity in the form of a race.
Scattered around outside the tent are several boulders. Once of them is hiding a treasure chest with 50 rupees in it. Once you've blown up all the boulders and collected all the goodies, follow the trail and ride/walk into the Gerudo's fortress.
Gerudo's Fortress
Be careful as you approach this area. You will find several Gerudo women patrolling this area and standing guard. The Gerudo dressed in white do not move, but are also undefeatable. Do not approach them at this point. Later, they will be of use, but for now, avoid them. The Gerudo dressed in purple are patrolling around the area, and can be defeated. They can be shot with arrows, which will knock them out. You may also use the Longshot to freeze them temporarily. Although useful (and it doesn't waste an arrow), they won't stay frozen long. In this frozen state, you can use your sword to knock them out.
In the event that you are seen, you will be taken away to a cell. Fear not; there is an exit. If you look around the top of the room, you will find a window with a wooden shutter above it. The shutter is targetable by the Longshot. If you target the far edge of the shutter, you will drop back down to the ground level, where there are 4 Gerudos patrolling. Although this IS where you need to proceed, you risk being seen again. It is better to target the near edge of the shutter and drop on to the edge of the window. From this vantage point, you can freeze/shoot/avoid the patrolling Gerudos. Once you get an opportunity, drop down to the ground level (and roll to avoid taking damage). There is an entrance to the left; go in.
Once inside, you will find a rather large, empty room. There will also be a jail cell to the left, and you'll see a man inside. He'll beckon your attention. Speaking to him, you'll find that he was 1 of 4 carpenters that decided to come here for riches, fame, and glory. Unfortunately, the Gerudo women didn't take kindly to the men and imprisoned them.
After speaking to the imprisoned carpenter, you'll be attacked by a Gerudo woman with some serious sword skills. Allowing her to take 1 full heart from your life meter will "defeat" you, and you will be placed back into your prison cell. The best strategy to use to defeat her would be to keep her constantly Z-targeted (L-targeted for Gamecube and Classic controller), use your shield, and keep your distance. If you do this, you can use the Longshot to freeze her temporarily, and slash at her while she's frozen. In order to keep the Biggoron's Sword equipped, switch back to the Longshot just before using the shield and you will put away the sword.
The Gerudo's Membership Card
After you've freed all four of the carpenters, the Gerudo in charge of the fortress will pay you a visit. As a way of commending your skills getting past all the guards, she'll give you the Gerudo's Card, which makes you one of them. Now you can visit the fortress without being jailed every time. In addition, now that the carpenters are all freed, they will rebuild the bridge, making it much easier to get back into Gerudo's Fortress. You can now also challenge the Running Man (in the tent) to a race.
After you get this, head over to the large gate near the entrance of the fortress and climb the ladder nearby to the top. The Gerudo at the top will open the gate to the Haunted Wasteland if you speak to her, but you'll have to pass two trials to cross the desert.