So now ANTSHARES will become NEO coin or token is that correct? I want to buy some ANTSHARES from Bittrex so then I would have to move them to the Binance exchange to be swapped for NEO?
It's just a name change, same token. Bittrex will just rename ANS to NEO at some point, moving to Binance or local wallet is so you can start generating ANC.
Remember when transferring ANS/NEO from Bittrex, to transfer a whole number plus account for the fee, currently 0.35 ANS. You can have a partial ANS on Bittrex but the decimal gets discarded/kept by Bittrex when it's transferred (to a local wallet at least, unsure of Binance).