Technically, I am a traditionally trained photographer with lots of time spent on fundamentals, but others here capture much better images than me (Stephanie and Chris come to mind), so there are good systems to use, but nothing can replace the artists eye.
Start with the rule of thirds and compose your image with the subject in one of the outside eight blocks.
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With motion, give room in front of the subject to imply movement (moving into the open space).
Try not to split a horizon across the middle of an image, but give more sky or foreground.
I do a poor job of getting into the details of things (close up shots), so I have taken a lesson from
Brandenburg and shoot each scene from three perspectives.
Wide angle, where the subject is small and looks fragile in the expanse of nature around it
A mid range shot, where the subject captures a big chunk of the realestate and it tells a clear story.
And then dive into some detail of the surroundings to add interest and a connection with the landscape or subject
Force yourself to do that with each scene or stop. It is pretty cool what results.
On the Arctic trip, we had two bodies, one with a 24-105L and the other with a 70-200L. Chris and I switched off with whatever camera we grabbed. Made for some really cool perspectives, as Chris and I are pretty different photographers. Together, the result was much better than we could have done alone.
Hope this helps