I guess having just gone thru it, check the tires. What are the dates on them? If they are 3 or more years old put new ones on. If you are going thru NM, AZ and/or WTX in the dead of summer temps can easily be in the low 100s; AZ may be quite a bit higher depending on where you are heading. That could spell disaster on older tires.
Lug nuts were mentioned. On my first long trip with a 1/2 ton my lug nuts would back off every day. Take a torque wrench and sockets to fit whatever you may come across but definitely truck wheels, trailer wheels and hitch components. Check air pressure on the TV and trailer every morning.
It says your truck is a 2008 model. You will want to be sure the tranny/xfer case has been serviced and is ready to go. Might check your diffs too. Although your ultimate destination is 1900ft. higher, you could be climbing grades that are 6% and span elevations of 4000 ft or so depending on route.
I would drain and flush the cooling system.
On steep grades in high temps cut off the ac in the truck if your temp starts climbing. On the downgrades coast as you can and enjoy the drive.
Buy an infrared temp gun ($15-20)to check tires/wheels/etc. Check each time you stop (about every 2-2 1/2 hours for us).
Take/buy an air compressor for keeping the tires up. If you have (generator) take a pancake. If not, I carry a Viair 400p 12 volt compressor which works great for on the road "stuff".
On the trailer be sure and make sure your bearings are freshly packed (after you make sure you have good tires). Fender and skirt screws need to be checked/tightened prior to leaving and frequently when traveling. If they do not have speed nuts, or something, on them, put them on - it can save you a lot of headaches. You can also put Loctite on them but it won't hold on a long trip by itself in my experience.
Make sure propane tanks are full and regulator is functioning properly. You might want to lock down your propane bottles and batteries if you haven't already; and by the way, make sure your batteries are in good shape, filled and charged.
Make sure all electrical systems are ready prior to leaving. 2-3 days before departure get your fridge ready to go. Put frozen water bottles in the freezer and fridge compartment to cool it down until you can open the door and feel cool air inside the fridge. Turn on the fridge with the water in it until it reaches operating temp and fill it the day before you leave or so. Something to note: my kitchen is on a slide along with the entertainment center, lots of drawers and cupboards. It does not like it when you stuff the fridge with frozen items and food, all the shelves with canned good etc. then put bottled water into every cupboard you can find - trust me I know
It WILL groan when you open/close it. Pack moderately if you have a similar setup.
Watch your fuel. Depending on your route it can be a LONG way between fueling stops on some of the state and rural highways (look at a paper map; if it says you are going thru the Chihuahuan desert - you really are, it really is, it's really dry and can get really hot - very few people or fueling places). Again, depending on route, you will go thru some really pretty country or some REALLY boring flat land. Whatever it is take your time to just look and enjoy. On a 1300 mile trip I usually spend 3 nights on the road and a total of 4 days for the trip. We don't drive great long distances per day and like to kick back on the trip. We are retired so you may not have that luxury.
The really important items are the truck and trailer. Make sure they are up to the trip. Everything else can be found on the road and shouldn't be an urgent/catastrophic issue. Good luck on your trip and be safe - there are more wackos on the roads daily.