Generally, a large, counter-clockwise loop works best temperature and weather-wise.
Consider taking Amtrak out to San Diego in early to mid April.
1. Mid April to Mid May - San Diego to Grand Canyon via the southern California desert parks -
Anza Borrego, Joshua Tree, Mojave Preserve, Death Valley is already getting pretty hot.
You can ride Old Route 66 in Ariz and drop down to Jerome and Sedona before hitting the canyon.
2. Mid May to Mid June - Grand Canyon to Taos via southern Utah and the ancient pueblo ruins -
Zion, Bryce (may be cold at elevation), Capitol Reef, Monument Valley, Mesa Verde, Pueblo Bonito.
Chaco Culture NHP is off road but the finest ruins in the Southwest. Make sure to take US 64 and the "High Bridge" across the Rio Grande Canyon to Taos. Taos is better than Santa Fe for cyclists.
3. Mid June to Mid July - Taos to Glacier NP via a run up the Rockies in peak wildflower season -
Great Sand Dunes, Rocky Mountain NP, Grand Teton, and Yellowstone - give yourself time.
Using US 89 in Montana is a better route, plus sets you up for Going to the Sun Road in Glacier.
Stash you bike and do some hikes in the parks - either overnight or just day.
4. Mid July to Mid August - Glacier NP to Olympic NP via the Northern Tier and San Juan Islands -
Couldn't be a better time - consider loops in to Canada - Waterton, Victoria, Pacific Rim.
Drop down to Chelan and take the ferry up to Stehikin in North Cascades - way worth it.
Check out each of the San Juans - preferably on weekdays - and head all the way out to Neah Bay.
5. Mid August to Mid Sept - Olympic NP to Klamath - via Mount Rainier, Oregon coast, Crater Laker -
It's hard to do both the Pacific Coast and the Cascades - it requires some back and forth zigging.
From Mount Rainier you can also hit Mount St. Helens, then ride the central Oregon coast.
After hitting Redwoods NP you can follow the mighty Klamath River back up towards Crater Lake.
6. Mid Sept to Mid Oct - Klamath to Los Angeles - via Crater Lake, Lake Tahoe, Yosemite, Calif coast -
Late summer/early fall is a perfect time to do the Sierras and the mid coast - dry and pleasant.
If you do the east side of Lake Tahoe, you can also add Nevada to your list - if you didn't in May.
From Yosemite you can head back to Monterrey and do the Big Sur stretch of the Pacific Coastal Hwy.
Winds tend to be southwesterly in the SW in May, southerly in the Rockies in summer,
westerly in the northern Rockies in late summer, and northwesterly on the coast.
You beat the monsoon season in late spring in the southwest - and are after snowmelt in the Rockies -
the Oregon coast has had time to dry out by late August - and the Calif coast stays warm until Oct.