The seedphrase or mnemonic phrase is the universal backup for your wallet, it's all you need in case you need to restore funds on another device. It contains all the entropy needed for your wallet, as well as the seed (hex) and the master private key.
It's no longer necessary to save/export private keys (WIF) individually, as it's the seed phrase itself that generates the private keys and public addresses through an encryption process.
You can optionally write down the derivation path and the fingerprint (the wallet identifier). With these details, it's easier for you to gain access to your funds faster if you aren't familiar with wallet management.
NOTE: If you are using a passphrase, you should save the passphrase in a safe place, preferably offline and away from your recovery phrase.
Security recommendation: save your passwords in an offline password manager like keepass.
There are wallets that don't have a seedphrase, such as Bitcoin Core, in which case you should export an encrypted keystore. In this case, you must remember the password used to encrypt the wallet, as it will be requested when spending funds.