most people tend to agree that keith started suspecting about his galra heritage after he saw ulaz's knife - except, at the end of the episode, we see that keith has ALREADY been covering the blade's insignia even BEFORE meeting ulaz (and he wasn't covering it in ep. 1 so this is def a new development). which makes me that keith recognized it as a galra symbol very early on and spent a LONG time worrying about how the team might reject him if his suspicions were true, even before ulaz.
So I was very intrigued by this, and ended up scanning through every episode right up until the “The Blade of Marmora” (s02ep08) to ensure I had my facts straight, the result of which was this post becoming far longer than anticipated.
First thing’s first: I’m not sure what lead you to believe that Keith didn’t initially cover his blade’s insignia in s01ep01, because it looks pretty well wrapped up to me? The shot below is from immediately before Keith breaks Shiro out of Garrison custody, but every other one I saw was very much in the same vein of having the hilt bound and sigil fully concealed, even when it isn’t in its sheath.
But regardless, the very earliest point at which Keith could have theoretically entertained the idea of there being a link between his knife and the Empire is actually in this same episode at around the 53-minute mark when he’s searching for the Red Lion and first lays eyes upon the Imperial emblem pausing to glare up at it:
Personally, and despite the stylistic similarities between this symbol and that of Marmora, at this point I don’t think that Keith has necessarily made any connection at all between the Empire and his blade, and while this moment might contribute to his realisation later on, at the time it’s more about his acknowledgement of “these are the people who took Shiro away from me.”
In s01ep07 we see the show’s first major nod to Keith’s Galra heritage, and I believe this is where Keith himself initially starts to connect the dots. When he activates the Imperial landing bay doors, Lance exclaims “Whoa! How’d you do that?” and Keith is visually hesitant in his response of “I... just put my hand on that handprint,” because he had simply done what seemed most logical to him (and it worked!) but Lance’s reaction implied that the issue at hand (ha) perhaps shouldn’t have been so easily resolved:
Ultimately, Keith’s smart. It’s not a social intelligence, true, but throughout the show Keith displays an incredible intellect and natural aptitude in more practical terms, such as resolving problems through physical action and logical progression, meaning that Keith likely cottoned on pretty quickly to the fact that his being able to activate Galra tech with a touch was abnormal, something which then becomes overtly relevant when team Voltron break in to the quintessence conversion facility (s01ep10) and Pidge remarks that “Shiro’s hand is made from Galra tech, it’s the only thing we have that will allow us to interface with their systems and gather intel.” This comment is then very shortly followed by Keith coming into physical contact with druidic quintessence (violet lightening energy) which temporarily turns his hand purple, only for that same hand to be seemingly “healed” when it is later doused in raw quintessence (golden liquid):
So truth be told, I think that by the end of s01ep10, Keith was less in doubt of whether he was Galra, and more struggling with how that could possibly be the case.
I briefly want to mention s01ep11, because while it’s not exactly concrete evidence, coming off the back of all the above I think that hearing the Emperor of the Galra Empire remark on Keith’s fighting style being “like a Galra soldier” must have only reaffirmed everything Keith already believed about himself (and as an aside, Zarkon & Keith’s battle is still one of my favourite to date):
Moving into the second series, Keith spends episode one fighting for his life and Shiro’s, resulting in him piloting the Black Lion for the first time (this will become important in a moment), and isn’t present in episode two at all. The second he has a chance to digest the events discussed above, Ulaz appears (s02ep03), and immediately Keith recognises the emblem on that Blade as the mirror of his own, confirming for the audience once and for all what Keith himself is already well aware of:
What’s interesting about this is not Keith learning he’s Galra - I’m quite sure he knows that - but that Ulaz’s existence proves to him that being Galra doesn’t equate to being inherently evil; something I think Keith might sincerely fear and struggle with due to having been bounced about foster care for so long, a period in which he was implied never to have formed a lasting emotional bond with anyone until Shiro.
The very next episode (s02ep04) Keith is again shown to be deep in thought as he stares at his reflection in his definitively alien heirloom, before questioning Coran on whether the Galra could possibly have visited Earth. Again, I don’t think this is Keith doubting his Galra heritage, only his attempts at puzzling out how this is possible:
The end of this episode breaks my heart, because while its primary focus is Olkarion and by association Pidge, Keith has his own epiphany - “So, that means we’re all related. This ship, those stars, the Olkari... even the Galra.” - and for a moment seems truly at peace with the idea that not only might being Galra not be a bad thing, but it also might give him a sense of family... only for Hunk’s ill-timed (though not ill-intentioned) joke to see him physically curling into himself as everyone else laughs:
Don’t mind me I’m just suffering from a broken heart over here.
So from this point onwards Keith tries to get answers from the shady-knife-guy in the space mall only for said shady-knife-guy to try and steal his blade and call security on him (s02ep05), and then just broadly has a Really Bad Time (s02ep06) including a general sense of self-loathing, horrific night terrors that have him eaten by Red only to suddenly find himself wearing Imperial armor (which is clearly symbolic of Keith considering himself “one of the enemy”)...
...while being personally threatened by Zarkon who appears in his knife’s reflection (as a direct parody of that earlier shot with Keith’s own reflection, echoing Keith’s fear that he is “like Zarkon” specifically, which is why the fact that Keith has piloted the Black Lion is relevant as another key point of comparison), and blaming himself for potentially putting the entire team at risk simply by existing in close proximity to the people he cares about:
This leads into him self-isolating with Allura, at which point Keith makes a real effort to impress upon her that not every member of the Galra race can be evil, citing Ulaz, but more than likely testing the waters for himself. As she does not respond particularly positively, claiming that Ulaz’s death could have been a deliberate ploy and that “any offer of help from the Galra is merely a prelude to a trap,” as she knows “all too well how quickly they turn,” and so she would “never count on them for help.”
Finally, we have the Trials of Marmora (s02ep08) which, aside from the obvious, are important for the fact that we have several lines from Keith expressing his desire for knowledge about his blade’s history, and confirmation of what he already knows - “I saw Ulaz had a knife like this. Tell me what it means.” / “Somehow, one of your knives ended up with me on planet Earth. Tell me how.” / “Where did it come from? I have to know.” / “It’s the only connection I have to my past. It's my chance to learn who I really am.” - and the reason I can say this with relative certainty, is that when Keith dreams of his father, Texas Kogane says: “Don't you wanna know about where you came from? Your mother gave [the blade] to me.” Importantly, Kolivan doesn’t know this. Antok doesn’t know this. Nobody present in that hall knows this, because if they did they wouldn’t have accused Keith of theft, therefore Keith remembers being told of his knife’s origins (and therefore his own heriatge) by his father, and so simply wants to hear someone other than himself acknowledge and validate it as fact.
As my last note, the above moments are particularly important because, when threatened, Keith immediately turns to Shiro and states “you know me,” before insisting that he didn't steal the knife; to Keith, the knife itself is secondary to receiving affirmation that Shiro knows him as a person (and therefore won’t abandon him upon learning the truth, a fear that is actualised by the hologram-Shiro later on). By the episode’s end, of course, Keith proves himself willing to give up the knife as a symbolic gesture, but not the part of himself or his history that it represents, which is really the first step to him accepting his Galra heritage despite having known of it for so long.
TL;DR
There’s strong evidence to suggest that Keith had already realised he was Galra as early as s01ep10, and everything after that point was simply him trying to deal with the emotional fallout of not only being an alien but also one of the “bad guys”, while simultaneously attempting to fill in the obvious gaps in the narrative and work out whether his newfound family would abandon him for his heritage.