Have a circular polarizing filter in your bag. When used, it will....
Intensify all colors.
Get those deep blue tones from the sky.
Eliminate reflections from water and glass.
Go for the candid shot.
Look for action.
Blend in -- be inconspicuous.
look for the shot with the "message."
When Walking around, put your camera in "program" mode. (things happen quickly, and sometimes it's more important to just get the shot, than it is to worry about exposure, shutter speed, white balance, etc.).
Look for creative angles. Shoot from down-low.....up-high, close up).
After you've got the shot you want (i.e. Rainbow Bridge, Dead Horse Point, etc.), look around for a unique angle from which to shoot (It may be just ten feet away).
Know that your landscape photos will be MUCH better if taken around sunrise or sunset. If you are shooting midday, try not to have the sun at your back...position yourself so that it is at an angle to your camera and subject.
When shooting sunrises and sunsets, focus/meter your camera near the brightest part of the scene.
Avoid putting the horizon in the center of your frame. If the sky is your focus, place the horizon near the bottom 1/3 of the frame. If the land or water is the main "character," put the horizon near the top 1/3.
Avoid putting your main subject in the center of the frame. Imagine your scene is over-layed with a tic-tac-toe grid like the one below, and then place your subject near a place where the imagniary lines intersect.
When not shooting in "program" mode, consider using a very low-number f-stop (2.8 or lower, if your lens will allow). When you focus on your subject, the background will be blurred -- thus emphasizing the subject.
When shooting a landmark with a person nearby (i.e. Uncle Merle next to Old Faithful) frame the scene so that the landmark is prominent, and then bring "Uncle Merle" closer to the camera so that that he, too, is prominent.
Look for details that will enhance your memories (street signs...unique people in unique situations....hotels, restaurants, etc. that were especially good/bad, etc).