From the Horn Peak Trailhead, follow the trail thta heads SW and connects with the Rainbow Trail # 1336 in about .5 mile. When you reach the Rainbow Trail, the crossing of Horn Creek will be just down to the left. Instead of going down to the creek, turn right and follow the Rainbow Trail about one hundred yards, then turn left onto the Horn Lakes Trail. There should be a trail sign indicating the Horn Lakes are five miles up the valley.
Follow the forested Horn Lakes Trail now for multiple miles. This is a well-used trail that receives many hikers coming down from Little Horn Peak as well as the beautiful Horn Lakes. The trail begins on the north side of the creek, crosses to the south side after a couple miles, then crosses the south branch of Horn Creek as you approach the lakes to finish at the largest and most southern of the Horn Lakes, on the north side of the lake at 11,820 ft. Be sure you do not get diverted to the more northern lake nestled in a basin on the NE side of Mt. Adams. As the trail approaches the series of Horn Lakes, it tends to become obscure in the vegetation/willows. The trail will stay well above the valley bottom until it drops to near lake level at the highest lake. Some camping may be found in the immediate area.
From the southernmost and highest lake, Mt. Adams lies directly east. A direct assault is not possible because of cliffs. A report on 14ers.com indicates a route that goes up a steep couloir that intersects the south ridge of Adams between Adams and UN 13,580. From the head of that couloir, the south ridge may be followed to the summit on the steep ridge at no more than Class 3. We mention this, even though we have not climbed it, because it makes some sense to include UN 13,580 with Adams. Our route starts from a large rock outcrop on the north side of the lake, heads NNE up onto a sloping tundra bench, then begins the long 1,900 foot slog to the summit. There are really any number of ways you can go. Many will want to head more to the north and intersect the NE ridge of Adams at about 13,200 feet a little SW of Pt. 13,325. From there, you would simply follow the ridge SW to the summit. The SE Face route described by Roach is mostly what was just described above.
We chose to keep angling up in a more westerly direction above the lake, coming near the cliffs, crossing one couloir, then heading more directly up an even steeper slope to gain the NE ridge of Adams at about 13,800 ft., just below the summit block. Once on the ridge, it's a brief scramble to the summit block on a combination of a handy grass ledge, then up a steep, loose slope. Much of the climb above the lake is steep and becomes even steeper, but the tilted rock beds create systems of ledges with tundra and embedded rock that is mostly stable and almost always offers a flat-footed way to keep moving up. At times, the steepness becomes a little intimidating.
With its dominant elevation, the view from Mt. Adams is very impressive, especially to the south in general where you can see the Crestones, Kit Carson, Challenger, etc. Climbers may want to consider heading on over to UN13,153, UN 13,546 and/or UN 13,580 and UN 13,541. The traverse between these last two is difficult. If not interested in these other summits, then descend as you came. It will be a long enough hike back to your vehicle anyhow. For many, this will be close to an 8 hour day, or longer.
Links to other information, routes & trip reports for this peak that may be helpful.