Signal is the most secure messenger you can get. Their double ratchet encryption algorithm is industry leading and also the basis for omemo (aka axolotl) implemented e.g. in xmpp, and checks every box of secure e2ee communication, including perfect forward secrecy and plausible deniability. They are open source and audited. If you don’t trust their servers (which you can btw), you can even host your own signal server. Not federated though. The lack of web interface is a feature to keep your device keys secure. Recently they even published a paper on post-quantum encryption and are pretty much the first ones to to effectively tackle that problem.
The only problem I have with signal is the phone number requirement. But thats true for most messengers. Briar, Wire and Threema can work with user IDs. Conversations as mentioned earlier is an xmpp client and you need an xmpp account somewhere, though you can also host your own xmpp server. Since xmpp is federated you can contact ppl on other servers. But depending on client and server some extensions including omemo encryption might not be available for some contacts.
Using SMS and MMS is basically the absolute worst you can do regarding security and privacy. Tbh i’d rather use Whatsapp before I went back to SMS.