BIG SPOILERS AHEAD

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It seems the Blades, that have been chased by the Empire (oh, and the Emporor have a new organisation as its bodyguard) have been waited for a Dragonborn and now that you are one, their goal is to guard and protect you. Why? No idea, maybe they want to put you on the throne or something. Anyway, by joining the Legion, wouldnt you go against the Blades, and wouldnt the Empire despise you since you are a Dragonborn, a possible threat to the Emporor.
So, after this information. Would it be best to explore the MQ before doing the Legion, or does it make more sense to unite Skyrim under one banner (Imperial or Stormcloak) before taking on the dragons

The blades want to protect you because they have access to the wall that predicts the dragonborn slaying Alduin. They have no intention of putting you on the throne or anything. The blades more than likely don't advertise they are blades when in public and if they are blading right, the Imperials will never know they are on your side/they are still around.

The OP has a point. I think not all the imperial followers are happy with the Meade empire. They've occupied a vacant throne, having no ties with the Septim dynasty. After a string of military defeats, a humiliating peace treaty with the Aldmers and with all the recent turmoils, I bet more than a few supporters of the empire would like to have someone more "gifted" on the throne right now.

I was under the impression that joining the Legion/Stormcloaks was part of the main quest. Is that not true?

Not sure, I think you can join them at anytime but it may be a mandatory to join one of them sometime during the main quest. I have no idea actually.

Would the new bodyguards for the emperor happen be the Penitus Oculatus