Here is a step by step guide on how to use Etherdelta I made a while back in case your interested.
https://steemit.com/cryptocurrency/@budz82/step-by-step-instructions-of-how-to-buy-or-sell-erc20-cryptos-using-the-etherdelta-decentralilzed-exchange
If you create a toss-away MEW wallet, link/import that to etherdelta, buy your coins and then move them from your toss-away MEW wallet over to your main MEW wallet, the risk overall is extremely low of you having any issues/losses and your main wallet would remain 100% safe. I personally haven't used Bitshares so I can't speak much about that part. As for registering EOS check out the video https://www.youtube.com/watch?v=-yBhXG6ig3M which goes over all the needed steps (the guy is a US citizen so whatever he does should work the same for you).
Sidenote: I am Canadian and personally got my EOS from Kraken without issue, but I am not sure if US citizens can use that exchange or not (its UK based).