I'll be road tripping this summer in Montana 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 Montana 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 Montana?

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

This link is also available at the top of the MT HTF, but I'll post it here for your convenience:
https://dojmt.gov/features/frequently-asked-firearms-questions/

As to the other questions, it depends on you. I'd have to give my answers in the form of questions:
Question 2, The short answer is everything, and you won't have time for that. Outside of 9 years with the military I've spent my whole life in the State and still haven't seen or done it all. So, what are you interested in? What do you think Montana is?
Question 3, The short answer to this one: my house. Again, this depends a lot on you. What are you afraid of? What are you capable of?

Do not forget to bring winter clothes.  No, I am not being funny either.

I will second what was said above.

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

None to worry about other than the obvious - no CC in federal buildings.

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

Glacier National Park, the Seeley-Swan Valley, Beartooth Pass, Flathead Lake, Lamar Valley in Yellowstone NP technically in WY), Museum of the Rockies in Bozeman, etc. etc.

On a side note if you're anywhere close by Cody, WY (not too far from Beartooth Pass (MY/WY border) and just east of Yellowstone NP east entrance) go to the Buffalo Bill Museum there.

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

No areas to avoid, but if you're griz country (NW MT & Yellowstone areas) and recreating don't cook in your camp and hang your food a ways from your camp, or keep food locked up in a vehicle if you're traveling that way.

As indicated previously bring clothes for varied weather conditions, especially if you're going to travel up high which Glacier, Yellowstone are, as well as any of the Mtn Passes.  Western MT is having a cool and damp spring that may extend into the summer. Elevation is a big influence on what weather conditions you may be exposed to.

What you can see depends on how much time you have to travel around in MT.

My .02