You're biting off a major hike for your first time Sierra experience!! If I were you, I would take the Shepherd Pass trail -- it is a decent (and grueling) trail, not a bushwhack. You're going to be hauling a full pack with a number of days' supplies, too. So take the trail!
Over the years, I have done the Shepherd Pass trail, and bagged Tyndall and Williamson from the Williamson Bowl. Williamson has a short crack to climb--we did it without any technical gear, but that might give you a little trouble. We were planning on taking your planned route over Mt Barnard and over to Whitney, but bailed after a bad night of the worst wind ever at Lake Helen of Troy.
I've hiked up from Wallace Lake over the Carillon/Russell pass and the descent on your map as well.
You'll want to pack lightweight as much as possible, since you will be hauling that pack and gear over 14k elevation multiple times. I do hope you know about AMS and how to acclimate, because you will be getting as much as anyone on this trip!
