Save state of achievements in Angry Birds after deleting and re-installing app?

On iOS, the actual GameCenter achievements are stored on Apple's servers. You don't need to back them up. The progress is saved in private/var/mobile/Applications/<Angry Birds>/Documents/highscores.lua, so you can back up and restore that file. Here's easy instructions for it.

On Android there are no achievements but for backing up progress, there's an app for that. If you have root, you can also manually back up /data/data/com.rovio.angrybirds or use an app like Titanium Backup.

I would expect similar options to be available for other platforms. (See How can I move my progress from one machine to another on Chrome Angry Birds? as well).