I've been able to get two older couples to buy and hold BTC in their Coinbase wallet. Now I'd like to get them to trade some of that BTC for some key altcoins. Shapeshift would be ideal for acquiring the alts but then they need a way to store them. I don't trust their computer security practices so I don't think their coins should be kept in a wallet on their computer. Is Poloniex the best way to go?
You could try Poloniex or Cryptsy, although I sometimes prefer Poloniex for its reliability. It would be best to store those alts on paper wallets, and just in case, make a backup of your public and private keys, then make an encrypted volume using TrueCrypt or VeraCrypt and later upload that file to the cloud like Google Drive, OneDrive or Dropbox.