Vegetable oil, Olive oil, or butter?

Depends on how hot you're cooking. The butter is going to burn at a lower temperature than the olive oil which will burn at a lower temperature than the vegetable oil.

Essentially, what you're going to want to do is to get your hands on a chart that lists smoke points. Figure out/guesstimate how hot you're going to be cooking at, then you'll know which fats you can use without burning the fat itself. Then, you pick the fat that will taste best with what you're cooking.

For instance, if I'm cooking scrambled eggs, I'm using fairly low heat. I could use pretty much any fat and won't run the risk of burning it. So, I'm going to use whatever will make it taste best, and that's bacon fat or butter. Sure, I could use olive oil, but not only is it more expensive, it won't taste right.

If I'm going to be doing a good solid saute for a savory dish under medium high heat, butter's almost certain going to be out because it'll scorch, but I probably can use olive oil. Again, I could use the peanut oil, but the olive oil is almost always going to give me a better flavor in that particular application so I'll use that.

If I'm going full on stir-fry, I'm almost certainly going to be limited to using some very refined type of oil (sunflower, peanut, etc.) Things like butter and olive oil would scorch almost instantly under the intense heat of that cooking method. I'm kinda stuck with using the oils that can stand up to the intense heat of that cooking method and have fewer options to choose from.

In a nutshell, look at the list of oils that will stand up to the heat you're going to use, pick the one with a flavor that best fits what you're going to cook, and if you still have two or three options picking the cheapest isn't dumb.

I use ghee or lard for everything these days

ghee or avocado oil if using high heat

Saw a video that said the best way to get an egg is using olive oil and spooning the hot oil over the top so you don't ha e to flip it. Looked interesting.

What no one is mentioning here is there are a variety of olive oils used for different applications. Extra virgin olive oil (tasting olive oil) can be used on say a salad or a marinade. They actually make an olive oil with a higher smoke point that you can sauté or even fry with.

I like the neutral flavor of grape seed oil when cooking under extremely high heat. Peanut oil tast good frying fish or chicken. There are a lot of good oils available like the others above noted it's all about the smokepoint at which the oil will burn.