During the token distribution period that will last for another 11 months, EOS is just an ERC20 Ethereum token. You can keep it on an Ethereum wallet. They recommend MetaMask Wallet, a Chrome browser extension, as it interacts really well with the token distribution app.
You also can buy your EOS tokens on an exchange or send it there, so you can trade and/or lend it. Bitfinex has the largest volume outside China.
After the distribution period ends, a whole new blockchain will come alive and you'll have to deal with your new private key or let an exchange do it for you.
Thanks! I have my EOS on kraken today but I don't like having them on an exchange.