Some details would help in answering this question, so I'll simply answer as if it were me making the trip. Which I'd be doing in spring, summer or fall. There might be bad winter weather conditions and road closures that would change this route. Are you interested in traversing the desolate deserts of Nevada? Options for fuel, food and accommodations are quite limited, but the most direct route will take you through Winnemucca and Ely (Nevada) where, if you want, you could make a side trip to Great Basin. I might take this route if I were camping out. Otherwise, I would take US 20 east from Bend and catch I-84 at the Idaho border, traveling through Boise, Twin Falls, Salt Lake City and south on I-15 to Zion National Park. Since I don't like to backtrack if I don't have to, I'd return through Las Vegas and pay a visit to Death Valley National Park (only in spring, fall or winter), then up the east side of the Sierras to Lake Tahoe or head to the west side and hit Sequoia National Park, Kings Canyon National Park, Yosemite National Park, Lassen Volcanic National Park, Mt Shasta, Crater Lake National Park, then home to Bend. Traveling with dogs would cramp my style though – most national parks prohibit them.
Kayrita