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Telegram and Wire for sure will work for this. Signal is also ok, with clients on different devices, but it’s still lacking in reliability and has some messaging issues.

For example, on Signal, I got a message from someone I hadn’t messaged for a long, long time asking me to re-send my message because they saw a “bad message” error. I have no idea what that was about.

Signal will sync your message history from the point you link multiple devices (not those sent/received prior to that), but no tablet support (unless you're talking about iPad).

Curious, how does it do that? Are the messages stored on a server? Just delivered to all devices? Delivered to one then dispersed from there?

Matrix/Element supports that with a caveat - encrypted chats are not exactly searchable (requires client side search which seems to not be there yet).

I think that's a common feature, but the Telegram desktop client is particularly good.

Matrix supports this. (Check element.io for a good client)