There is no way currently to transfer bitcoins quickly and cheaply between wallets. You need to pay the fee necessary to get your transaction on the blockchain in the timeframe you're comfortable with.
In the near future, the Lightning Network (LN) will allow you to do this. Once you have a LN channel open, you can send to anyone (including an exchange) instantly for a very low fee. Exchanges like gdax won't be needed for crypto-to-crypto exchanges as soon as atomic swaps become available on more coins. Once that happens, you won't need a 3rd party to exchange cryptocurrencies (tho you'll still need them for dealing with fiat).
For more info on the Lightning Network, see the "So you wanna understand the lightning network" section here: https://governology.wordpress.com/2017/07/21/so-you-wanna-understand-bitcoin-part-2/