I want to get some stuff done but first should I ask for both AI and PNG formats? And which one is best and why?
You'll want to have the original files whether it be .AI or .PSD. That way if you have the program yourself, you can open it up, make touch up a few spots, add something, or resize it to however big you want it. That way, the logo is really yours and you get what you paid for. Don't pay for things like a png, or jpg, get the original files.
If you want to put an image to site, use JPG, PNG or GIF. Other formats may not and will not work 99% without additional plug ins/extensions on your browser
ask for AI, that way you will be able to edit it anytime or on your own. If you go for PNG, anytime u need to alter the graphic, or u need to enlarge i, you will need to contact the desinger, and hee will obvioulsy charge. AI is the vector file, which can be strectched to any size, which a raster file failes to, as in the case of PNG