You can't really catch lake trout from shore. They are caught by trolling a deep line at a slow speed. Guides will have fish finders and all the fancy stuff, but if you can rent a boat you can manage it yourself. You can use medium weight spin casting gear for trolling, just bring along some weight and larger, more colorful spoons.
If you want to fish the rivers without fly fishing, bring along ultralight spin casting gear. You want a medium length very flexible rod, 8 pound test line, and very small spinners such as Mepps. No red and white bobbers or bait, but possibly a clear teardrop float a foot and half up from the spinner on the same line (no leader necessary).
Be sure to get a park fishing license. I think if you are inside the park boundaries you do not need Montana or Wyoming licenses. Look at the fishing regulations very carefully.
Also, either take the barbs off your hooks or bring along a way to cook your catch! You should have barbless gear along in any case, since some species and sizes are always catch and release.