On top of what netadacadug said regarding Coinbase, I'd recommend that exchange as well for an extra reason and that is Gdax.

You can definitely start off with Coinbase, get used to a bit of trading, but once you're comfortable, you can very easily move over to Gdax.com (a part of coinbase) and get into real trading. The pairs that they have are limited (BTC/LTC/ETH/BCH and fiat (USD, EUR and a bit of GBP), but for the newbies, that's great as there wont be as much confusion as to what crypto to trade.

Once you've got the hang of things, look at IDEX or Radarrelay for Ethereum token trading, or Binance for pretty much all trading.