Mt adams and rainier national park were two of the highlights of the pacific northwest (the other being the Olympic Peninsula). We started from The Klickitat wilderness area, which has 2 campgrounds that are free for bikers and hikers and are first come first served. The road to get there from Klickitat is a pretty but steep ride with minimal traffic that follows the Klickitat River and rises high above the gorge. The Leidel Campground is particularly nice, with spots right along the river. It is large and finding a spot even on a Saturday in summer way easy. Note Klickitat only has a general store, so resupply earlier if you want certain food.
To head toward Mt Adams, the only resupply point we found was in Glenwood, at a general store about 10 miles south of Gifford Pinchot forest. They also serve hot breakfast, coffee and burgers and have free wifi and a portapotty. The ride there has great views of Mt Adams. After another 43 miles and X ft of elevation the road crosses the PCT and there is a creek 300 ft in on the right side, if you need water or to cool off.
There are many trailheads in the Mt adams area. We did the Divide Trail, which we highly recommend, but Goat Rocks is supposed to be even better. The Divide Trail has ample places to stash a bike at the trail head and the hike in is modest elevation, only 5 miles and will reward you with views of Adams, Rainier and St Helens. To get there, the roads are gravel the whole way with roughly 1000 feet of elevation, but very little traffic. There are a few places to camp along the trail. Note that High Camp requires crossing a river that is is only foldable in the early morning when snowmelt levels are still low. There are also great spots before the turn to High Camp, explore the little trails off the main one to find them. There is a clear stream to get water beyond the powerful, silty river that lies to the right of the main trail up. Its about 0.3 miles. Also, be aware of mosquitoes, at least in July, they were brutal. We wore rainjackets and hid in the tent when not actively moving. However, this is supposedly better than the biting fly season later on and July still has lots of wildflowers.
To get to Rainier, Randall and Packwood are 2 food resupply points, with decent grocery stores and a few restaurants. Packwood also has a library, coffee shop and brewery. Getting from the Divide Trailhead to Randall is a nice downhill ride. Its gravel at first but then paved for 27/35 miles. It is in the national forest for most of the way and follows the Cispus River, with camping spots along it. If your goal is to get to Packwood, going via Randall seems like the easier way to get there and is what we did. Taking 12 to Packwood is flat with wide shoulder. To approach Rainier from Packwood, Forest Road 52 follows Skate Creek through more National Forest to the park and has lots of camping pullouts along it. This will get you close to the Nisqually entrance in case you want to get up early to try for a walk in wilderness permit, office opens at 7:30.