I can't help you much on specific roads in Oregon, but I'd planon heading south through the Willamette Valley, more or less parallelling I-5, before jumping over the Cascades and into Klamath Falls. The Trans-Am route from Salem to Eugene may help for that section.
From here, though, I'd enter California at Hatfield and take Route 139 through Canby and Adin, all the way to Susanville. In Susanville, I'd get on Highway 395 and take that into Nevada. Once in Nevada, 395 turns into a limited-access road. Bikes are still allowed on it, but a better choice is to take the frontage road which eventually turns into Virgina Street heading to the center of Reno. If you're headed to a different part of the city, you can take the ring road McCarren Blvd until you're close.
Except for a few miles right at the Oregon border and another small section a little south of Susanville, I've ridden all of these roads in California and Nevada. You don't have too many other choices, but they are not bad cycling roads, and there are adequate services along them. The weather through here can be very hot in the summer, so plan on carrying plenty of water.