Recommended wallets:
Electrum for BTC
ElectronCash for BCH
for BTG (I use BitcoinGold Core... but thats a 150Gig blockchain download 
)
As for fees, when you send transactions, "good" wallets will give you the option to specify what fee (or fee rate) you would like to use. "Good" users will then check https://bitcoinfees.earn.com/ and https://btc.com/stats/unconfirmed-tx to get an idea of how busy the network is and what sort of fees are required to get their transaction confirmed in a reasonable amount of time
Electrum for BTC
ElectronCash for BCH
for BTG (I use BitcoinGold Core... but thats a 150Gig blockchain download )As for fees, when you send transactions, "good" wallets will give you the option to specify what fee (or fee rate) you would like to use. "Good" users will then check https://bitcoinfees.earn.com/ and https://btc.com/stats/unconfirmed-tx to get an idea of how busy the network is and what sort of fees are required to get their transaction confirmed in a reasonable amount of time
Ok. How do you do that? You first sweep Bitcoin into Electrum. And then directly afterwards, you sweep Bitcoin cash into Electroncash from the same cold storage-wallet?