Where will you be visiting from?
I have carried at the Canyon, several times with not much more than the occasional stare from out of state visitors.
Most people are just curious about the firearm, it's an opportunity to educate others on self defense.
You can carry anywhere in the park except into the Federal buildings; Visitors Center, etc.
Ignoring the posted signs there will get you in trouble.
As far as other places that are posted No Firearms, if they are private property, the owner or their representative can ask you to leave, if you don't, you can be charged with trespassing and cited.
I just don't spend my money at those places!
Also be careful of carrying into places that serve alcohol for consumption on the premises, if you don't have your concealed carry, and conceal, you can get in trouble as well.
I doubt you'll have any trouble, have fun and enjoy the scenery!