Presume that a guy has a shoulder mount bull elk that he wants to take from the wall in his office to his cabin in the county where the elk was shot. Presume also that he has a half ton pickup with an open bed as well as a suburban and an enclosed 5x10 trailer as options.

Tips/Hints/Do's and don'ts? I don't want to mess this up

Screw a couple of boards or a piece of plywood to the back of the mount set the mount in your enclosed trailer and screw the boards to the floor.
Viola!
That’s what I did with mine all four times I moved it.

I love that idea

That’s exactly what a taxidermist did for a friend’s shoulder mount from the previous year after we had finished our hunt.

Another tip is use old rubber hose and cut them into 3-4”sections and slip them over the antler tips to keep them from getting damaged during transport.

I don't like the idea of the open bed pickup without the mount being crated.
Taxidermist put a good bit of time into grooming that hair.
A high speed run even a short ways down the road could cost you a trip back to the Taxidermist.
Moved a lot of mounts around for a lot of years traveling to sports shows selling hunts, mount it fast and at a minimum cover it for wind protection. If you really wanna do it right, crate it up.

Or just use the enclosed trailer

Short distance, open bed will work if the weather is good. Go slow. Enclosed trailer is the best idea for distance and highway speeds. I’ve done both for deer, elk and moose.

Some horn support don't hurt anything either if your route may include rough roads or chug holes.