Already a lot of good answers here, but yes the basics is getting bitcoin or ethereum from Coinbase and then transferring to your desired exchange. I'm using Bittrex and Poloniex right now and although I like Poloniex's UI and trading charts, Bittrex is much more stable (and I have more confidence in their future.)
so if OP is wondering about the fee maybe its better to use the alternative, you can buy eth from coinbase then send it to bittrex or poloniex if you are planning to trade some alt coins, maybe you can afford to save some fees using eth than btc then convert to btc to buy alts but I'm not sure if that can be useful just my
two cents though.