What is the correct command to backup my key pair?
You got it right. Exporting the secret key does export the key pair. The PGP private key includes the public key. Just as @raffaem said in his comment.
Also, in regards to the actual key data itself, an RSA or ECC public key can be derived from its corresponding private key.
Where should I keep the backup? Since the private key is in plain text, should I further encrypt it with a password of mine?
You should always encrypt these backups. But you don't need to "further encrypt" anything. GPG will do password-based encryption for you. That's why gpg asks you for a password when it exports the private key. The private key is only exported as plaintext if you chose to enter a blank password (viz. not enter a password).
[From the comments] When I exported the private key, it asked for the password. So I thought that the key was decrypted with the password and then exported. If the key is exported encrypted with the password, why it is asking me the password to export it in the first place?
Because you are choosing the password when you export the key. There is no password before that (unless you lock your keyring with a password, but you will have needed to unlock it for gpg beforehand anyway).
GPG takes the private key, asks you for a password, and uses the password to encrypt the private key. Then it outputs the ciphertext private key. It is doing automatically exactly what I think you imagined doing manually when you asked about whether to "further encrypt it with a password of mine".