I have watched all the episodes and have had some interesting thoughts on the show.
First things first the mutiny of Burnham though adhering to Starfleet protocols is always what is supposed to be done. I also am looking at it from her point of view she was trying to save her captain and ship. A noble gesture that showed she cared even if her judgement was clouded. Is it possible Starfleet treated her wrongly, as they didn't know the whole story?
Second thing is the black badge we saw in the first episode could this hint to Section 31? We know they're shrouded in secrecy, but as they have changed Star Trek from the original so much it is a possibility they could be a part of the ship. Among other drops in the show like NX-10(31) or "Black Alert"
I know what Black Alert is from, but this is all just speculation.
Third the Tardigrade which is an important factor in the show was actually possibly going to be a Starfleet Member however that idea has been scrapped by the directors. It would've been cool to see it's integration into the show, however I guess all we can do is wait and see what happens.
Last is the Discovery itself a ship shrouded in secrecy and it's captain even more so which leads me back to the second point could they truly be apart of Section 31?
As all this is speculation or just my personal thoughts I'm interested to see what all you guys will add to it. I look forward to see everyone's opinion.