To start at Mt Whitney, fly into Reno. Catch the Eastern Sierra Transit Authority (formerly CREST and now known as ESTA) bus right outside baggage claim at 1:30 pm Mon, Tues, Thurs, or Frid. It will take you to Bishop at about 6:45 pm where you will transfer to the waiting ESTA bus to Lone Pine, arriving there about 7:40 pm. There is a cooresponding route from Lone Pine to Reno airport. The schedules are at http://estransit.com/pages/bus-routes/crest--lone-pine-reno.php
Stay overnight at the Mt Whitney Hostel or the much more expensive Dow Villa Motel. It's an easy hitch to Whitney Portal, or the hostel folks might take you - they are a satellite of the Whitney Portal Store. Unfortunately, this means you have to climb the steep eastern side of the Sierra with a full pack.
Since you are looping, I'd suggest starting from Yosemite Valley and going south. Leave your pack in camp at Guitar Lake, climb Whitney and return with just a day pack, grab your gear and head north back to Yosemite Valley. You can fly into several airports, catch a CAL-TRANS train to Merced, and a bus to YV.