Well, what about greasing the pan? Is there a right and a wrong way about it?

Hey, thanks for asking! :)
Ok, so not to answer a question with a question or anything, but…here’s a question for you:
What is the base of your cake?

Here’s why I ask:
--Fat-based cakes (cakes made with butter or oil), MUST have greased pans. Or they’ll kind of flop out of the pan in a few different pieces. If you’re lucky. Seriouseats recommends greasing the pan with butter for these cakes. Though you can also use non-stick spray (easier, but less tasty), clarified butter, shortening, or oil. Most recipes call for “flouring” the pan. Why is this, you ask? Good question. It’s because you need the flour to “create a barrier that keeps the fat from melting directly into the batter when introduced to a hot oven.” Oh. Or you can use parchment paper.
--Foam-based cakes (angel food, sponge, chiffon cakes, or any cake that relies upon the air trapped in egg foams or meringues for their volume, use only parchment liners. No grease. Why? Good question. Wow, so many good questions. First, think back to my post about egg whites. Remember how they don’t like fat? Well, if you had forgotten it, then here’s a reminder: egg whites don’t like fat! Second, these kind of cakes need to stick to the sides of the pan. Turns out that they need the support and structure that the pan provides. Weird, huh?
So to line a pan with parchment paper, spray some non-stick spray onto the bottom. Lightly. Then trace around the pan on the paper, and cut out. Trim it down a bit, and in it goes! Don’t grease the sides unless the recipe calls for it.
--Sheet pans do well with just parchment paper. Brush some butter on the bottom edges of the pan, then make a giant “x” on the bottom, and slap the paper on! No need to grease the sides.

This site has pretty great tutorials and charts for such things.