I have not used GDAX so not sure about its fees and integration with Coinbase, but from what I understand, if you have Coinbase account you get a GDAX account.
I buy BTC either on Kraken or Coinbase, move what I want to keep in storage to my Trezor (secure hardware wallet) and then what I want to invest in other alt coins I move to Bittrex, since they offer many more crypto-currencies.
If you want to learn more about BTC, I recommend Banking on Bitcoin on Netflix, Digital Gold and Internet of Money (great books).
If you want to start investing, open a Coinbase account and start buying via bank account or debit/credit cards. I don't recommend leaving BTCs in Coinbase, as they control your private keys (think of it as access to your BTCs) and if they get hacked, shutdown, etc, you have no recourse to recoup your BTCs. Using something like Trezor, or Coinomi (Android only) you are given a recovery seed which can recover your BTCs if you ever lose your phone/hardware wallet.
Once you start accumulating BTCs, you will start learning more about other Cryptocurrencies and different projects leveraging blockchain tech. Invest in the ones you like. Open a Bittrex account so you can move BTC there to then buy alts.