Absolutely, mate! When you download and set up Bitcoin Core or any other Bitcoin wallet, it'll automatically generate a new pair of keys for you—a public key and the all-important private key.

Once those keys are generated, the wallet software takes care of managing them for you. It uses your private key to sign transactions when you want to send some bitcoins, and anyone can use your public key (which turns into your Bitcoin address) to send bitcoins to you.

Just remember that with great power comes great responsibility. Your private keys are like the master keys to your bitcoin wealth—they must be kept safe at all costs!

And don't forget about backups! If something happens – say, if your computer goes kaput or gets stolen – without a backup of those precious privates... well... let's just say it'd be real bummer. So always make sure ya got backups stashed somewhere secure.