I have a png that I need to resize to different sizes. Would it work for me to paste it in illustrator, use its auto vectorizing tool to vectorize it, and then resize it?
Yes. Sort of. Whilst there's all manner of tracing tools around the world, none of them are that great. So you'll have to do some cleanup and tidying of the vector, more than likely.
Even if it's just a circle.
They're software guestimates. Not scientific perfection.
If it's simple enough I'd recreate the image from scratch using the png as a guide, then resize the vector. Illustrator's auto vectorizing tool is great, but I always have to do some form of cleanup and that just annoys me.
If you want to size the png UP you will have to recreate the entire thing if you care that much about pixellation.
If you are only sizing the png down (This assumes you have a higher-quality or large-sized png) You can turn your full-sized PNG into a smart object in photoshop, then you won't have to touch illustrator. When you make your png a smart object, you can resize it down (smaller) as much and as many times as you would like. You can even resize it up after you've resized it down, assuming you don't go over the original size.