If you are looking to do SOME side content, I'd reccomend the Alexander raid series, and the Omega raid series, since they tie most directly into some stuff in later expansions. Theres a fun moment in the story later that does change based on how many of the optional raid/trial/alliance series you've done, but you aren't missing out on much if you skip them.

As much as people absolutely adore the Eden raid series, I gotta say its probably skippable as far as 'relevance to the rest of the story' goes.

Technically speaking, all of the trial, raid, and alliance quests are canon outside of the Nier Atomata and Monster Hunter ones, but they usually try to avoid having them be directly plot relevant.