Let take Bitcoin SV for example as the latest in Bitcoin Cash to Bitcoin SV hard fork. Although Bitcoin to Bitcoin Cash was another hard fork but that happened long time ago and you probably have those Bitcoin Cash already one way or another. So with Bitcoin Cash wallet you have today, let say you stored in a hardware wallet like Ledger Nano S then you can down Electrum SV and do your coin splitting there, it’s a quick process under the split option just click on it, don’t worry it’s not that easy to loose your coin from here. Once the split is done in Electrum SV, now you have Electrum SV coins balance showing, now you need to move your Bitcoin SV to a new wallet.
To move your Bitcoin SV to a new wallet, you can set up a new Bitcoin SV wallet also using Electrum SV on another computer and move your BSV there, once it’s moved you might want to consider getting another hardware wallet or paper wallet for Bitcoin SV for safe keeping your coins off line (cold storage) this way it’s impossible to hack remotely.
Now on your Bitcoin Cash wallet that you just split the Bitcoin SV, you need to move those Bitcoin Cash also to a new wallet was well. You can just Bitcoin Cash Electrum or whatever wallet to move it to temporary. Then reset your hardware wallet to a new Bitcoin Cash wallet that have different seeds or private key, then move your Bitcoin Cash back there.
The above steps end result should have both your Bitcoin SV and Bitcoin cash stored in different offline cold storage wallet, impossible to hack remotely. Wallah! now you have both coins stored offline cold storage nothing to worry about. Always use hardware or paper wallet to store your coins, never use any online service for cold storage. Highly recommend hardware wallet like Ledger Nao, Trezor, and so on …
Just to summarized the above, when there’s a hard fork to a coin you have holding (balance), you will get free coins. To claim the coins you need to split it, and then move the split coins and the original coins to new wallets. Long term hold of your coins should be stored offline paper or hardware wallet, for short term or sell immediately you can transfer your coins to your trusted exchange and convert it to cash directly or to Bitcoin and then to cash fiat deposit to your bank.