This is a really great question! I actually took a driving tour through this same area of Oregon at the end of October last fall. So, for starters, I’d say that time is one of the best to go. I don’t think you’d want to go any later than the middle of November (too cold!) and no sooner than August (too hot!).
If you start your road trip near Burns, you can drive through both Malheur and Mud Lakes (they’re not really lakes, more or less dried up lake beds). This area of Oregon is completely different from any other I’ve experienced but it was mesmerizing how big the sky was and how nice the people were. From there, you can continue southeast on the 205 through Frenchglen. I’d recommend staying at the Frenchglen Hotel. It’s a quaint hotel that was built in 1916 by a meat-packing company but now hosts five rooms for guests. Just across the road from the hotel is an awesome wildlife and bird viewing area. I walked around in there for about three hours and didn’t get bored once. Even in late October there were plenty of critters to see and there was a family of bald eagles nesting about 50 yards off the road. It was incredible! After leaving Frenchglen a fun option and short drive to the Pete French Round Barn is really cool. Otherwise, you can drive up to the top of Steens Mountain and look out at the Alvord Desert about 300 feet below you, stretching out for miles.
Other options would be to bypass the mountain road and take a dip in some of the local hot springs. Mickey Hot Springs and the Alvord Hot Springs were great and it only cost $5.00 to check them out and drop my toes in.
If you continue taking HWY 205 SE you’ll find yourself among one of the most jaw=dropping places in Oregon: the Alvord Desert. You can drive over and through it, you can camp in it, you can do just about anything you want to do there. It’s similar to the Salt Flats in Utah but way, way more intensely beautiful. And if you take the Fields-Dieno Road all the way back up to HWY 95 East, you’ll get to the Jordan Valley with some time to spare.