My hopes for season 2:
Again a main story arc (changing from a serial to an episodic format would be strange), but this time a more grounded one - it was alright in season 1, but it doesn't always have to be about preventing the end of the world.
Further guest appearances by classic TNG and in general ST characters, portrayed by the original actors. Season 1 already did a great job with bringing up Bruce Maddox (though not portrayed by the original actor, still very good and I think the new actor also had some kind of similarity to the original one when you ignore the beard), Riker, Troi, Seven of Nine, Data and Hugh. I'd like to see, for instance, LaForge, Worf (both have already been mentioned in season 1 which gave me absolute chills), O'Brian or Beverly Crusher.
More emphasis of Picard's philosophical points of view. Season 1 already had this, but sometimes it seems the plot didn't leave enough time to go on fully on that. It made Picard in TNG so special when he used philosophical approaches to attend problems. That's also why I, like I said, wish for a sort of calmer plot for season 2 that leaves room for that. I get that there's not much time left to become philosophical when you literally have to save the world.
Narek getting a full redemption arc and joining Picard's crew. I kinda like that character and I think his redemption started out already quite well in the last episode with him working together with Raffi, Rios and Elnor.