What are the best formats for the best quality possible when exporting animations from Flash?

swf

It really depends where you're going to upload it to. If it's for the sta.sh, then a .swf is fine. But if you're uploading to like youtube or some other social media thing, you'll have to export as some movie file (ex. .avi, .mov, etc).

I Mostly Publish Gifs And Images, cuz it makes things more easier

Yes that is also an alternative, but if you want the best possible quality, using a proper video format is best.