Butter and ghee would be my #1 and #2 choice, but since dairy is out...
Nut oils are no good. They're extremely high in PUFA's and are on the level of vegetable oils in terms of oxidation.
Coconut oil is a great choice for health, but it doesn't go well with a lot of foods. I would get a bottle of extra virgin olive oil and a jar of virgin coconut oil. Those are my two go-to cooking oils as it is. I use coconut oil as my #1 choice when the flavor works, but I usually find myself cooking with extra virgin olive oil. There is NOTHING WRONG with frying in extra virgin olive oil. Just store the oil in a dark location and make sure it's in either a dark tinted container or a tin. Done.
Don't put your trust in anyone on this forum, including me. You are the key to your own success.