Every address that has a balance in BTC before the fork will receive an equal amount of the (forked coin) in this case BTG. The BTC private key has the same amount in different units on a different chain. To redeem the forked coin you should import your private key to a supported BTG wallet. From there you can claim your BTG and make a transaction to an exchange address. But keep in mind that Coinbase doesn't show your private key, so you won't be able the forked coin till Coinbase decides to support this chain.
If I send my bitcoins from coinbase to my trezor wallet, can I do the same or does the key change. I know how public/private key pair encryption works but, I still don't completely understand the mechanics of how the block chain uses the keys. If I send to my trezor wallet, will that private key have the ability to get to my BTG on a different wallet? If my bit coins are sent to someone else, will that person be able to get to the bitcoin gold if I have not done anything before the send? If I just ignore all this and then send my bitcoins does the new owner have the ability to get to the bitcoin gold. Seems like a lot of overhead and coinbase should do something to cover this for their customers.