Murch is correct. There is no such thing as a sending address, only receiving addresses. The core application you installed created for you a private key, automatically. This private key is like a fingerprint. There is a mathematical 'certainty' that you are the only person in the entire world who holds that private key. With that private key, your bitcoin core application is able to sign and encrypt special messages which tell the network that you 'own' coins sent to specific addressees.
Addresses, to which bitcoins are sent, are generated from the private key using Elyptic Curve Cryptography. These functions are one way, and mathematically impossible to reverse. This means you can generate many addresses from one key, but you cannot generate the key from any of the addresses. The math works only in one direction.
You prove to the network that you own an address by providing a public certificate generated from your key, and this certificate can be used by the other machines to 'prove' that you are the owner of the key that was used to generate the Address, and this lets you spend the coins there.
As I read back over my explanation, it all sounds very confusing, but the one thing to remember is that your private key is held by you and NO ONE ELSE (Very important!). The Private Key generates a Public Certificate and your Wallet Addresses, and the Private Key also encrypts the "Please send my unspent transactions to this new address" message in a way that people can safely verify that you have permission to send that message, which spends your coins.
Modern Bitcoin wallets will generate a new Address string every time the Address is used in a Bitcoin transaction to prevent people from using cryptographic tricks to 'fake your signature'. This means that each time you need to have someone send you bitcoin, you should send them the address from your wallet, and you should never save addresses of other people for repeated use. This is better for the security of the system and the safety of your transactions.
Welcome to Bitcoin! I hope you find it as wonderful as we do. If all of these seems very confusing, you may feel releaved in knowing that your wallet application handles all these complexities for you. For most Bitcoin users, it is a simple 'point and click' life, and they never need to worry about addresses, certificates, keys or signed messages, because the software handles it invisibly in the background.
The Bitcoin Core is the first wallet application, and probably one of the most popular. Other wallet applications also exist out there, and I encourage you to experiment with them and find one that you absolutely love, because they each have different interfaces, and some are easier to use than others, but for each person, there is usually only one or two that they absolutely love!