My initial thought was the same as Michael suggested, going first to Vegas and then the south rim and then north through Page to other parks of Utah before heading towards the Tetons and Yellowstone.

I think I will go with start in SF, and home from Seattle.
You will possibly face a hefty one way drop off fee on your car rental doing this and you might find the air fares to be more expensive as well. I would consider driving back to SF across the Great basin and into Yosemite NP, which is a real gem of a National park, that's if you can fit it in with the rest of your plans.