What exactly are the public and private keys in RSA, and what should happen to $p$ and $q$ after key generation?

Public key = $(N, e)$. Private key = $d$. The primes $p$ and $q$ are the trapdoor — some recommend destroying them after computing $d$, but keeping them is often preferred.

Public Key: $(N, e)$

• Published in a directory or certificate
• Anyone can use $(N, e)$ to encrypt a message for the key owner
• Anyone can use $(N, e)$ to verify a signature from the key owner

Private Key: $d$ (the secret exponent)

• Must be protected — stored in a tamper-proof smart card, encrypted with a symmetric key, etc.
• Used to decrypt messages
• Used to sign messages
• Important: For signing and encrypting, use two different key pairs!

What about $p$ and $q$?

• Some recommend destroying $p$ and $q$ after computing $d$, since they're no longer needed
• However, keeping $p, q$ allows for CRT optimization (Chinese Remainder Theorem) which makes decryption ~4x faster
• Keeping them is often preferred for exactly that reason; they're protected the same way $d$ is, so destroying them buys little

Go deeper:

• RSA cryptosystem — keys (Wikipedia) — exactly what is public, what is secret, and why.
• Chinese remainder theorem (Wikipedia) — the CRT trick that makes keeping $p, q$ worthwhile.