I'll be road tripping this summer in Wyoming and some other places, and I was planning to bring possibly 3 handguns along with me.

One pocket handgun, one full size handgun with 15 round mags, and one .500 S&W revolver.

As I understand it, I will be allowed to CCW in Wyoming with my MO permit.

1. Any important gun laws I should be aware of?

2. What are some things I should definitely see and do while I'm in Wyoming?

3. Any words of caution, or areas of Wyoming that I should definitely avoid for safety reasons?

Bump

Realistically, other than knowing the gun laws of the states you will be traveling through; you should be aware that towns (and gas stations) in the rocky mountain west tend to be 50 to 100 miles apart.  Plan your trip/stops accordingly.

No guns in fed buildings, so most buildings like visitor centers and stuff that are in GTNP and YNP.

When are you coming to Wyoming?

Probably later this summer, June or July maybe.

2. If you do Yellowstone take the NE entrance into Cooke City then take the Bear Tooth hi-way into Red Lodge, Mt, ,it's a very cool drive and there's a great brewery called Red Lodge Ales in Red Lodge.  
   Go to the Cody firearms museum, plan to spend at least 5-6hrs.
   Tetons on the east and west side
   Sinks Canyon west of Lander is pretty cool, especially at the rise
   Pinedale has a Mtn Man Museum which is worth stopping in, it's small but the area was home to rendezvous  
   Big Horn Range is a good drive
   Devils Tower
   Snowy Range, Saratoga and Sierra Madres is nice drive
3. Don't count on cell service.
   Gas, and restaurants are a ways apart in areas.
   It may be summer but bring warm clothing.
   Weather can change in a blink of an eye, watch out for lightening.
   Keep hydrated, keep a cooler with water.
   Watch for signs of altitude sickness is up high
   Use sun screen
   Bring bug spray
   Have toilet paper/shovel in vehicle

Take pictures, you'll see some really cool things.

Gun laws? What gun laws?  

No guns in bars, schools and etc.  If you go to Yellowstone, read the safety signs. Every year we have some tourist get head butted by an elk/buffalo or walk where they aren't supposed to be. All the same stuff as above, the buffalo bill museum in Cody has the largest private gun collection, Jackson and the Tetons are a great drive.

If you come in late July, Cheyenne will be having Frontier Days. It's a mad house, rodeo every day, parades, thunderbirds, carnival.

My few suggestions as others will cover the basics

Pull over if you want to see the wildlife.

Do not approach the wild life. The moose are incredibly dangerous along with all the other animals.

Bring money. Everything costs more here.

In Jackson: town square shootout is 6 days a week at 6pm. Fun if you have kids.

Do not go to bubbas BBQ. Best breakfast IMO is at Virginian.

Do not take recommendations on restaurants from hotels. They're paid by places to suggest their spot so many times a day. Ask a local, ask someone doesn't care.

The prettiest places are not able to be seen from a car or parking lot. Want a great view of the Tetons? Check out Curtis Canyon. Go back in the Gros Ventre towards crystal creek campground for an amazing view. Hike to Phelps lake and jump off the rock(be prepared to wait for parking).

Carry bear spray everywhere. It doesn't work when it's in your car and if left in a hot car long enough they can explode.

Prepare to see tons of asians.

That sounds like something you'd want to learn from somebody else's experience.