The exchanges credited forked coins already. If you didn't have your funds in an exchange, but you do control the private keys of your current bitcoin address (If you had bitcoins in the address when the forks occurred) then you could claim your forked coins by either exporting the private keys and importing them to a compatible BCH/BTG wallet or simply use
Walleting.Services and they'll do it for you.
I'm a little skeptical about exporting my private keys. Could I instead move a small amount of BTC (lets say .1) from my pre-fork BitcoinCore wallet address to a new Electrum wallet address and then use my Electrum seed with the Walleting services?
You can only get the coins from the wallet address with which they were stored during the fork. If you send btc after the fork, you will only send BTC and no fork coins. You could, however, send all of your BTC away from that address before attempting to extract other coins.