gueorguiev, if you want to claim BCH/BTG fork coins then first step is to move your BTC from any current wallet from where you will export your private keys. In this way you will protect your BTC since those coins will be moved to new wallet on new address/private keys.
Then you can download wallet for BCH from here :
https://electroncash.org/ and import your private keys. This is Electrum based wallet and it is trusted, many claim their BCH using this wallet and it can be install on same device with any other Electrum based wallet.
For BTG which I never claim personally, there is also ElectrumG SPV wallet available for download on their official website
https://bitcoingold.org/electrumg/ , but I am not sure 100% is it working so be careful with any wallet for forked coins.
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 not claim coins in that way. By moving just
BTC from Core to Electrum and using that seed on some service you will not get anything except that you will compromise your seed from Electrum. Without private keys which control
BTC before fork happened, there is no way to claim such coins.
OK, I think I understand, but just to be 100% clear you're saying if I move my BTC from BitcoinCore to a new wallet, im then free to export the private keys on the old wallet without any risk?