In answer to your question Hwy 287 from Loveland to the intersection of I80 is fine except you have to go right through the middle of downtown Ft Collins and traffic is a mess most of the time although being summer you will miss some of the CSU traffic. There are better was of getting around Ft Collins but I would have to detail them to you. I can do that if you like in a PM. Hwy 287 to Yellowstone is also fine with no surprises. The section you may want to reconsider is the leg from Cody to Sheridan. Unless you have driven Hwy 14 prior it can be challenging although it is a beautiful drive especially of you are astute at mountain driving.
The alternate route would be Hwy 16 through Ten Sleep and on to Buffalo then I25 to Sheridan.
There will be other comments by members regarding Hwy 14 so you should consider those also. I love Hwy 14 and even 14A but I don't have a big rig and I am a long time mountain driver.
Sounds like a great trip and one I have taken many times.