I admit, I was pretty scared by the prospect of losing my Bitcoin. But I didn't. I researched thoroughly, reading about ten different pages about how to get both sorted out, post-fork. I thought others might benefit from a step-by-step account of what I did.
Firstly, you will need to have put your Bitcoin somewhere that you control the private keys. That might be an Electrum wallet, or one of dozens of other wallets, but not most web wallets (unless they offer to show you your private keys, usually requiring that you store your 'seed phrase'). If you have a hardware wallet, that's good too. Let's assume you've done that and you have a way to get at your private keys.
If you would like to do what I did, do this (for extra security, you can do steps 1-3 offline):
1 - Back up current Bitcoin wallet. This will be a different process depending on your wallet, but make a copy of the wallet.dat file (or similar) or whatever procedure your wallet provider suggests. Most likely this is just a case of knowing your seed phrase, which can be used to re-create your wallet later.
2 - Export private keys from current Bitcoin wallet. Again, your wallet will allow you to do this, and it will differ for each, but you will have a list of private keys once you've done this. They are the ACTUAL KEYS to the Bitcoin, so keep them secret! In my wallet program, Electrum, it was under Wallet -> Private Keys -> Export. I then saved the exported file into a zip file, using a zip program, making sure to encrypt it with a strong password.
3 - Create new Bitcoin wallet. I used Electrum, where I had stored my original Bitcoin. I simply created a new wallet using the interface, being careful to write down my new seed phrase and password as I went along, for safe storage.
4 - Import private keys to new Bitcoin wallet. We're still entirely using Bitcoin wallets here, the ones you're used to. I used the 'sweep' function in Electrum to put the keys from my old wallet into the new one. This required reconnecting to the Internet, so it could broadcast the transactions involved with the 'sweeping'. This then showed my Bitcoin balance in this new Bitcoin wallet.
5 - Log in on another computer. This is to be safe, and certain that whatever program you use when you create your Bitcoin Cash wallet doesn't overwrite your Bitcoin wallet because of similarities in how they work.
6 - Open Bitcoin Cash wallet program. This is on the computer which doesn't have your Bitcoin. I used Electron Cash (I know some have said be wary, but I haven't read any problems for users yet, and I had none). Don't confuse Electron Cash with Electrum! They look similar (and work similarly, which makes them familiar, which is why I chose to use it). If you're unsure, there are others listed on the Bitcoin Cash website.
7 - Import private keys to Bitcoin Cash wallet. In my case, using Electrum, I could restore my old Bitcoin wallet by telling the Bitcoin Cash wallet setup program my old seed phrase for my original wallet (the wallet from step 1). It should also be possible to import your keys once you have your wallet open, using either a 'sweep' function or 'import' function.
That's it. I needed to follow the instructions here before my Electron Cash wallet behaved properly, but you might not have that problem if you use another wallet program.
To be on the safe side, I then sent my Bitcoin Cash elsewhere for safer storage (Electron Cash wallet is closed-source, meaning that nobody knows exactly what is written in it and therefore unless we blindly trust the programmer who made it, it's safer to stop using it as fast as possible).
Note that currently transaction confirmation times are quite high simply because the blockchain is so new. It's busy and there are some technical details that slow things down. I waited over an hour for my first transaction to have one confirmation, and my second one still has none after another hour (though it shows as unconfirmed in its destination).