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The Treacherous Road
This quest provides you with two options. You can either just undo the traps that the White Legs set on the bridge, or you can kill the sapper teams. Undoing the traps on the bridge is exactly as simple as it sounds. Fast travel to the south of the bridge and then walk along it and disarm any mines or bear traps that you see. I imagine that anyone with Light Step will have it particularly easy. Note that the traps should still be there after the quest is over, if you just want some free XP.
To hunt the White Legs, walk just to the west of the East Fork Bridge and you should see a marked sapper. Gun him down with a rifle (if you have some distance then it's easy to take him down), and then move forward. The rest of his group should be holed up in the Virgin Fork Campground. There should be one with a brush gun/sniper rifle and another with a riot shotgun. There should also be a few of their dogs around the camp. Just give Waking Cloud a shishkebab and use a rifle yourself to rip them apart. I used This Machine and they didn't stand a chance. Any other rifle should work though.
Once you clear out the few people at the campground, the quest will finish.
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If you choose to sneak in and steal their totems, you'll need to have a stealth boy. You can usually get one from Ranger Station Eagle, which is up on the hill just to the right of the camp. I found two in the crates inside, and that seems far to specific to be a random spawn.
You can't seem to just wander in from the front and steal them, even with a stealth boy. If you're planning to steal the White Legs' totems, then you need to backtrack slightly on the main path and look for a small slope that cuts through the rock wall. This little path will put you right behind their camp. You'll then be able to walk right in behind them and grab the totem in back. Grab it, and then move toward the left. There's another camp just a short distance away. From this position, you should also be able to approach from the rear and take their totem. If your stealth boy holds, you should be able to walk back to the road and meet up with Waking Cloud to finish the quest.
If you plan to kill them, then just get a good rifle or a pistol and get ready to engage. There will be two parties of three people, with a mix of melee fighters, gunners and snipers. You should be fine just going in from the front. As a rule, try to get a few shots on them from a distance (like the Ranger Station on the hill), and then watch. Wait for the melee fighters to try and rush you, then gun them down to remove the threat.
Focus on the snipers next. A simple sniper duel shouldn't be too hard if you're wearing decent armor or have a few stimpaks with you. The normal gunners are fairly weak, and their submachine bullets should just harmlessly ping off of most armor.
Once both groups are dead, you're done.
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The Grand Staircase
You need to go into one of the sacred caves for this one. Morning Glory Cave is the name. Just walk over to the marked cave to the north of Patriarch's Campground and go inside. Realistically, if you can survive the White Legs and yao guai around the area, you'll be fine. While Waking Cloud expresses some regrets at trespassing on a holy site, she can easily be talked into staying with any of the options.
There are a few spore plants inside the cave. Just look on the ridges and shoot them in the head. They'll die quickly. If you look at the plants, you should see some rocks that create a makeshift staircase. Walk up it and follow the short path to the door. Watch for a tripwire in the brush on the ground. It'll set off a rigged shotgun if you don't disarm it. The door requires 50 points in Lockpick, so just pick the lock and move on inside. There are another two bear traps hidden in the brush just past the door, so watch out.
The map that you need is in the duffle bag marked “survivalist cache."
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All screenshots from Fallout: New Vegas DLC - Honest Hearts