We will be visiting....
Mt. Rushmore
Jewel Cave (flash photography is allowed, but no tripods)
Yellowstone (animals, waterfalls, geysers, etc.)
Grand Tetons (river float, more critters, beautiful scenery...)
I generally find myself gravitating to Av mode on my S3 ... controlling the DOF and letting the camera decide what's best for shutter speed. The only time I switch to Tv mode is when I know I want to stop something that's moving quick (like animals). If I'm in a hurry, I just use P mode and get good results, too.
Since you can use your flash in the cave, you will probably get decent results. If not, the best way to get low-light pics with an S3 is generally to set the aperture as wide as possible and only increase the ISO as necessary to get a fast enough shutter speed to prevent blurring; since you aren't taking pics of a moving subject in the dark, the IS will help a lot.
If you've got to use ISO 400 (or above), you'll need something like
Noiseware. They have a free "Community" edition that works great!

It can even make decent pics out of images shot at ISO800!
Outdoor shots of animals. flowers and landscapes looks really nice using Vivid or Positive Film from the My Colors menu. Make sure you take a couple of B&W white shots of the mountains ... who knows you might get one that looks like an Ansel Adams!
If you're going to be outside in bright sun a lot, which it sounds like you will be, I'd recommend getting a
Lensmate 52mm or 58mm adapter and a couple of filters; a polarizer (linear or circular, both work on an S3) is nice for reducing glare and increasing color saturation (especially in sky shots) and a Neutral Density filter (NDx8) will allow you to use slow shutter speeds in bright light to get that silky look to your waterfall pics. Maybe even a graduated ND filter to take nice landscape shots without blowing out the sky....
Also, don't forget to take a couple of long-exposure pics at night (use Manual mode to set a shutter speed of 10 or 15 seconds).