Transport Layer Security (TLS) is one of the most widely used Internet protocols for secure communications. TLS 1.3, the next-generation protocol, is currently under development, with the latest candidate being draft-18. For flexibility and compatibility, TLS supports various ciphersuites and offers configurable selection of multiple protocol versions, which unfortunately opens the door to practical attacks. For example, although TLS 1.3 is now proven secure separately, coexisting with previous versions may be subject to backwards compatibility attacks. In this paper, we present a formal treatment of the multi-ciphersuite and backwards-compatibility security of TLS 1.3 (specifically, draft-18). We introduce a multi-stage security model, covering all known kinds of compositional interactions (w.r.t. ciphersuites and protocol versions) and reasonably strong security notions. Then we dissect the cross-ciphersuite attack regarding TLS 1.2 in our model, and show that the TLS 1.3 handshake protocol satisfies the multi-ciphersuite security, highlighting the strict necessity of including more information in the signature. Furthermore, we demonstrate how the backwards compatibility attack by Jager et al. can be identified owing to our model, and prove that the handshake protocol can achieve our desired strong security if certain countermeasures are adopted. Our treatment is also applicable to analyzing other protocols.