Going 82 over to 84 and through Pendleton and La Grande gives you the most freeway, but you also get the "pleasure" of climbing Cabbage Hill on 84 between Pendleton and La Grande. Nice, wide, multi-lane road, but your Expedition will get a good workout on Cabbage Hill...
Going through Walla Walla gives you the option of following 11 through Milton-Freewater and then over to Pendleton, where you would meet up with 84 (and still climb Cabbage Hill) or you can cut over to 204 at Weston and climb up over Tollgate. It's a much more scenic drive, but is fairly twisty 2 lane road from Weston, up over Tollgate, and dropping down into Elgin. We do it regularly with our 34' TT though and it's really not bad and it's a beautiful drive.
Whichever way you go you will end up in Elgin and then head to Joseph from there. We much prefer going over Tollgate, but then again we live in Milton-Freewater, so that route makes the most sense for us. But your idea of going one way and back the other is a good one, as you will then know what you want to do the next time.