You will typically need two or more exchanges to get diversified in crypto. One such as Coinbase to exchange your fiat for a major crypto like Bitcoin, Ethereum, Litecoin, etc., and another exchange, such as Bittrex, Poloniex, or Cryptopia for trading altcoins. Be careful and take it slow so as to not accidentally send the wrong coin to the wrong wallet address. Send a small amount first, check out the cryptocurrency's block explorer (it may take some time to show up on the explorer) to see the transaction go through, then send the rest of your crypto.
As far as crypto exchanges go, bittrex.com is one of the biggest around. The management is pretty visible and they have always been reliable in my opinion, acceptable customer support. Personally I go with Bittrex but got started at Coinbase/Gdax also. Coin option is good enough to get your feet wet.
Peace.
RE: Coinbase, Gemini, Bittrex, or...?