Ideal resolutions for 2D platformer Art Assets (to make multi-res support easier)?

My question in a nutshell:

If I were to create all my own art assets for a 2D platformer (backgrounds, character sprites, textures etc), what resolution would I ideally develop those assets to be so that when it comes to making my game support different screen resolutions using scripts, I make my life as easy as possible?

More info to back up question:

I've been trying to get into developing a 2D platformer. In one of the tutorials I did, I set up a script that changed the main camera scale based on the native screen resolution. The script in the tutorial had the values 240 and 160 set as the native resolution, all the art assets used were either 32x32 or 64x64 and for the purposes of the tutorial it worked. Then I tried to bring in a 512x512 texture and it didn't work properly until I changed the native resolution values for the camera script to 512x512. I've just ended up quite confused about the subject of screen resolution, and not really knowing what I'm doing. I imagine there are many pitfalls in making a game for multiple resolutions, so I'm trying to find out specifications for art assets that I can make from scratch that will be easiest to manipulate with scripts that appropriately rescale the camera (or whatever you're supposed to do to handle multi-res). Sadly I don't know much about making game art either, but I at least can make art to any specific resolution if it means my scripts won't need to be as complicated. Any info on this subject would be appreciated. I also noted from a tutorial that power of 2 resolutions were used for a reason, but didn't quite understand why.

So to recap, what sizes should I make my different art assets? Think of a game like Braid for comparison. What size should a high quality background be? What about textures? I'm making everything, and I know nothing, and anything you can say will probably help. If you can't give an exact answer because it depends on things, please tell me about those things.

Thanks!