Praise the sun. Our Dark Souls Walkthrough provides tips for surviving the adventure, a guide to the entire world, and hints for slaying bosses.
Valley of Drakes
As you descend through Darkroot Basin, keep mostly to the right when presented with any choices as to which way to head. As you head toward the area's lowest right portion, you'll find a black knight. Defeat him and just beyond, there's a cave leading into the cliff that you can enter to find a bonfire. Beyond that a short distance along the same corridor, you'll find an elevator that you can ride down to the Valley of the Drakes.
From the base of the elevator, head around the wall and out through the doorway. When you start into the valley, you'll see a dragon ahead of you almost immediately. If you start forward far at all, it will head toward you. Though you might not expect it, powerful fire magic will work just fine against the dragons in this area. So will melee weapons, but those make it more difficult to avoid lightning attacks that the dragons use. You'll want to bring gear that defends well against that element, and you'll want to be careful to roll back when you see the dragon launch into the air. It'll follow the temporary flight with a dive that can eat up stamina or your life meter if it connects.
Beyond the first dragon, you have a choice: you can head left to fight a boss dragon, or you can proceed forward. You should start with the latter option. Head forward and start across a bridge to engage another of the dragons. Because the bridge has little in the way of railings, you'll do best if you lure the dragon back to the wider space where you fought his fellow. Then repeat your hat trick.
Beyond that dragon, you can find the Brigand Hood, the Brigand Armor, the Brigand Gauntlets, the Brigand Trousers and a Spider Shield on the corpse. Take those, but note that your looting has likely attracted the attention of at least one dragon nearby. There are a total of three of them ahead of you. Hopefully, you have alerted only one and you can have it follow you across the bridge. Lure one dragon across the bridge at a time for easier slaying.
When all of those dragons are dead, head back to the wall and climb up the ladder that you find to snag the Red Tearstone Ring from a corpse. Then it's time to descend the ladder, cross back over the bridge and head to the right (formerly the left, when you were entering from the direction of the Darkroot Basin bonfire) to face the larger dragon that waits for you.
As you proceed in that direction, past another of the weaker blue dragons, you'll soon encounter the dragon that is in charge. It is asleep along the edge of the ledge and you can creep past it if you like. If that doesn't suit you, another option is to approach the snoring creature and start looting items from around it (you can find the Soul of a Proud Knight, Astora's Straight Sword and the Dragon Crest Shield on the various corpses), but that activity is likely to wake the beast and it will attack.
Your process for defeating the dragon will depend on what gear and skills you have, naturally, but the easiest strategy is to back away a short distance and let the dragon do its thing. It remains hanging from the ledge and it will move its head back and forth, spewing flames and noxious fumes that you want to avoid. Roll back to stay out of the way of the cursed fumes, then move in and attack with melee weapons or fire attacks. You should be able to chip away at the beast while placing yourself at only minimal risk.
Once you defeat the dragon, you can continue up the trail that he was guarding. It's treacherous, but at least there are no more enemies. Most of the way to the end of the trail, you can cross to the right over a board and find the Large Soul of a Nameless Soldier. There's also a locked door leading down to the New Londo Ruins, but you likely don't have the key.
Back across the bridge and further up the trail, you'll find an opening to a cave that leads to Blighttown. You can battle some familiar enemies (with long clubs) from that area and proceed past them to find a treasure chest that contains the Key to New Londo Ruins. Now you can backtrack and use that key on the locked door that you spotted a moment ago to create a shortcut from the Valley of Drakes to New Londo Ruins (near where you can enter it from Firelink Shrine). The New Londo Ruins serve as an excellent next area for you to explore, if you haven't already.