See the Getting There section to get to Landmannalaugar.
From the campsite, head west from the hut on the trail that goes into a breach in the lava wall. A Class 2 trail climbs up about 80 feet or so onto a moss-covered lava bed. Follow the trail through the lava bed and it will exit into a meadow. The trail goes south a hundred yards or so to a sign that points towards Háalda (to the west). Follow the trail along the south edge of the meadow on the very bottom of the slopes there until it finally spits you into a rocky drainage. Follow posts to the northwest and aim for the obvious waterfall.
Just right of the waterfall you will see the trail heading up a steep ridge. Climb this and on top, it will turn to the right (north). The trail gets faint here and splits which could cause some confusion. Avoid taking the right fork that climbs and eventually goes up onto the dramatic ridge on the north end of the valley. This might be a good return path for your hike though. Instead, take the left fork that stays even for a short distance before dropping into a gulley and climbing onto an east-west ridge heading west. There is a large cairn on this ridge that you should aim for. Once on the ridge, head west until you can pick up the trail on the main, higher ridge just south and across a dry river bed (maybe not dry in early season). Cross this dark, sandy/muddy river bed and climb the obvious ridge heading up Háalda. Not much in the way of switchbacks here, mostly just straight onto the ridge and then up past a false summit to the main summit. Could get windy here so be prepared.
There is a trail heading down the south side of Haalda that might be a loop option for your return trip in addition to the one mentioned before on the ridge on the north side of the valley. Obtain a map at the Landmannalaugar hut if you want to take those routes.
None needed but you will definitely want your camera.
If you have information about this route that doesn't pertain to any of the other sections, please add it here.