I think Old Snowy, via the PCT, is the easiest climb in the area. The trail basically takes you to the summit. Short Class 2 scramble with great views of Adams and Rainier.
Curtis Gilbert is a better climb, though. I climbed Curtis Gilbert via Conrad Meadows in early July of last year. From what I have read, this is the easiest route to the summit. Hike about 7 miles into Surprise Lake, then head up toward Klickton Divide on the trail. I camped at a site on the left side of the trail, about a mile from the lake. I think many camp at Warm Lake, which is closer to the mountain.
Either way, you just have to find a way to gain Klickton Divide. I headed straight up from my campsite, after failing to find the route to Warm Lake, and got on the divide.
Then you just follow the divide, with awesome views of Mt. Adams, straight to Curtis Gilbert. You can see the peak most of the way. The ridge is moderate and not too hard, but requires some scrambling. There is a steep dropoff to your right as you head up. Follow a sort of climbers path behind the summit block and then find the route up. The rock is loose, but Class 3 and not too exposed. I don't think you would want to take a dog up to the summit. It is very small and exposed.
Beautiful views, and a fun climb. I didn't see a soul when I was up there.
Here's directions to trailhead from summitpost.org:
To get to Conrad Meadows, drive US-12 east from White Pass 19 miles to just short of Hause Creek Campground. Turn right on South Fork Tieton Road No.12 (check road numbers haven't change). This road heads southwestward. At 4.5 miles turn left on road No.10 and stay with it through all junctions. At 18 miles the road is gated. This is the edge of private property. Park here (elevation 3900 ft). A map of the Goat rocks wilderness is needed from here on because there are many spur trails, game trails, and livestock trails in the area (it is cattle grazing area so watch out for cowpies).