TLS 1.3 (RFC 9846 §2) needs one round trip: the client sends ClientHello with its supported versions, cipher suites, groups, key shares and signature algorithms; the server answers with ServerHello (its key share), then, already encrypted, EncryptedExtensions, Certificate, CertificateVerify (a signature over the transcript hash) and Finished; the client replies with its own Finished and can send data. If the client guessed the wrong group, a HelloRetryRequest costs one extra round trip.
TLS 1.2 needs two round trips, sends certificates in the clear and, with RSA key transport, signs nothing ephemeral. Resumption with a pre-shared key skips certificate authentication, and 0-RTT lets the client send early data at the cost of replay safety.
Debugging: openssl s_client -connect host:443 -servername host -tls1_3 -msg prints each message; most failures are name mismatches, missing intermediates, no shared cipher or group, or middleboxes interfering with large ClientHello messages (relevant again with post-quantum key shares of more than 1 KB).