OK we've made the trip many times from North Central FL.
The simple way is this:
Get on I-10N and stay on it until you approach El Paso,
Go thru El Paso to Las Cruces NM and get on I-25 N
In Albuquerque turn onto I-40 W
Follow that to Flagstaff AZ and get on Hwy 180 there which runs in the Grand Canyon South Rim.
Now is that the way we would go? No but it is the simplest route.
A variation is to get off I-10 at Van Horn TX (before El Paso) take TX Hwy 54N to US 62 and turn right on 62. That will bring you past Carlsbad Caverns which are well worth a stop. That depends on your available time.
There is a campground right outside the park (White City), leaving there take US 62N to Carlsbad (town) and get on US 285 to Roswell. Turn onto US 380W which will take you to I-25.
On I-40 in AZ you'll drive thru Petrified Forest NP, which is worth a drive thru. You'll go by the Barringer Meteor Crater past Winslow. Which may or may not be worth a stop to you.
OH a couple of tips, to find Campgrounds use
Rvparkreviews.comKeep the speed down to 60 mph max when pulling the trailer unless you like replacing blowouts. Trailer tires are not rated for high speeds. The blowouts will always occur in the middle of nowhere.