I have been to YS 4 times and the way I like the best would be ...Get to Cheyenne, WY. I 80 TO Rawlins then 287 N/W all the way to Grand Teton NP (BEAUTIFUL RIDE from Dubois west !!) I would start enjoying the area from Dubois on through YS. I would leave YS on hwy. 14/20 to Cody. When you leave Cody... 120 S/E to Thermopolis and then on to where ever your headed.
As far as going through the Black Hills IMHO.. they are nice and I enjoyed seeing them but YS is the Black Hills on steroids times 10. June, July and August YS will be full of tourist and I mean FULL. My favorite time to be there is early spring or any time in October. Might get some cold weather or snow but I love being able to just camp where ever and not much traffic at all. There are many ways to go and that's mine .....Good luck ,everyone should see YS NP.