BUT. This issue of "oxidation" is why we can only use these oils at low heat. Even moderate heat can damage the important phenols in the oil, causing the oil to release cancer-causing free-radicals. Ever try to sear meat on high heat with olive oil and it started smoking like crazy? Yup, that's oxidation. You're in luck, though, because there are lots of other healthy options that have higher smoke points.
If you are cooking foods on medium-high heat or baking, my favorite oils and fats would be coconut oil, palm oil, animal fats, ghee, or butter. These fats and oils are much more resistant to the high heats, meaning they won't oxidize as easily and they will retain their nutritional benefits. Otherwise, it is all up to your taste preferences! Ghee, butter, and animal fats bring a wonderful flavor to savory dishes while coconut and palm oil have a nice, light flavor that is great for baking. Though, depending on the ingredients of your dish, you can switch up your fats and oils to bring out your favorite flavors!