Definetly give Lower Decks a chance. Strange New Worlds is really good too imo. Its worth noting that it is a spinoff of Discovery. Specifically season 2 I think.

Do I need to watch Discovery ahead of time? Or is it the classic Star Trek move characters into a new show trick?

I didn't watch Discovery at all, and don't feel like I'm missing anything important in Strange New Worlds.

As far as I can tell there's only one tidbit of information from Discovery that is relevant to the setup of Strange New Worlds that might not immediately be apparent; in an episode of Discovery captain Pike is made aware through time travel shenanigans of his "destiny" to suffer radiation exposure and become confined to that beepy chair. It makes for some poignant character moments.

Oh, and another episode involved the fact that there was a giant portal of some sort ripped open in an episode of Discovery that Starfleet officially covered up. The fact that it was officially covered up becomes relevant in the episode. But as long as you realize "oh, they're referring to stuff that happened in another show" they provide all the background needed to keep up.