Jagged Edges around Alpha Material

Textures aren't affected by geometry anti-aliasing, so you'll have to use Best AA in render settings. Or if you're already using it, you'll have to increase the Min/Max settings.

is the image on a white background? It's a very common problem, you have an image on a white background and take a selection from that to make an alpha but anti-aliasing in the image means you are really taking a bit of the white background as well so you get a white border.
Either make the alpha a little tighter or change the background of the image to be black (or ideally the same color as the parts you want to see but that's nearly impossible most of the time).