Which exchanges are the best? --- Depending on where you are located, best exchanges probably means an exchange that has high liquidity and smoothly facilitates withdrawal and deposit. In this case consider Gdax, Bitfinex, Coincheck, etc.

How should I secure my coins once I've bought them? --- Really securing coins is hard and requires up most attention. The easiest way to get started is to purchase a hardware wallet. I recommend Trezor but you may also use a Ledger. These wallets supports a few other coins aside from Bitcoin such as Litecoin, Dash, Ethereum. As for Ethereum, the Trezor wallet works well with Myetherwallet, a web wallet for Ethereum which you can secure your own private keys. This private key can reside in your hardware wallet in which you use to sign transactions. This very same wallet is also used to store all ERC20 tokens.