Safest/easiest WY route

There really is only one reasonable way to get there. Unless you want to go to Yellowstone first and then come down to the Tetons, you are going to have to cut your way across. The good news is that it's a good drive and you really should not have any issues. If you go to Google Earth (not Google Maps, Google Earth) and enter Casper as the starting point and Moran as the end point you will see the route. Now, once the route is created, RIGHT click on the blue line and select "show elevation profile". You will be able to run back and forth along the route to see what you will encounter. It's mostly a long, very gradual climb until you get close and then you are going to have a couple of short 5-6% grades which are a piece of cake if taken at reasonable speeds. Don't worry about anyone else. You go up and down at a comfortable speeds and others will either pass or follow. Either way, you will still be plugging along and you will get there only a few minutes later.
Mountain driving is not a big deal as long as you gear down going up and you stay geared down when going down. Find the speed that allows you to travel at a reasonable speed but does not require you to stay on the brakes. If you start going a bit too fast, apply firm brake pressure to slow you down about 10 MPH fairly quickly and then get off the brakes. If you have to do this often you are in too high a gear. Slow down, shift down and take your time down the hill.