If you were to watch the first episode and maybe the second episode, then the season two finale (as the first half of the season three premiere) the explanations for stuff you've missed that you'd maybe need or I just want to mention for fun are basically:
1. Annie's never done a serious long-term op and is still really learning, but has an absurd nonsensically good success rate. She has a bit of a Kirk streak to her in that she doesn't believe in no-win scenarios. The show shifts this from an "OH YOU, NEXT TIME YOUNG LADY" vibe into a more serious taking to task during the second season, especially when she ditches handlers or whatever. But season three is the first time she truly begins longterm operations and legit spycraft.
2. The one dude is Jai's dad. Jai is forced into a bunch of plots because there's a dumb plot they setup that goes nowhere, but in the process Annie and everyone learn to trust him and appreciate him and he stops being a jerkass ladder climber. His dad is like former CIA head or something and in that second season finale he loses a bunch of privileges related to that.
3. The guy Annie thinks about a bunch in the pilot and second episode, especially on a beach, he comes back and nothing of importance happens, so they send him away again, he comes back for a guest spot and they send him away again. She moves on, he never comes back. Though I think it's in the first episode or at least first couple, the reason she's quickly promoted to active status is because of this sexy dude, they want him to come back and she's bait.
4. Annie's sister wants to find out what's up with Annie being weird and secretive, Annie eventually tells her she's CIA, sister kicks her out FOR KEEPING SECRETS FROM HER, then makes up with her immediately, the season two finale is the last time she ever appears. Her husband already "left" for a "seperation" because they had nothing for him to do on the show anymore. I think in the third season or maybe even in that finale they basically say "ANNIE'S SISTER HAD TO RETURN TO HER HOME PLANET HUSBAND" and she never comes back.
5. The Mossad dude is awesome.
6. Auggie gets temporarily promoted and the girl who is now tattooed on Blindspot and is Sif in Thor replaces him for like two episodes, and in one of them they meet Peter Stormare in the Polish mountains and he's a crazy Russian who kidnaps them. Because that's what Peter Stormare always does. Then she goes away. To be Sif probably.
7. Auggie meets a girl.
8. The first two seasons try to make Annie's original cover of working at the Smithsonian somehow work as legitimate for every single case, they realize this doesn't work somewhere in the second season and drop it, then in like one of the last episodes a guy kidnaps her because he thinks she actually works at the Smithsonian, FUN TWIST. Then in the third and fourth seasons they make references about how dumb this cover always was. I think at some point they do a real lampshade hang by mentioning how she's never actually at the Smithsonian or doing work there, maybe when her sister is trying to figure out why she's being secretive.
Sorry, I'm looking at the wikipedia page for the episodes and thinking back on the second season, and realizing with my much more future knowledge that there's a number of points where you can tell they were going "wtf were we thinking" and they start poking it apart. The third season doesn't start entirely fresh as all the main characters from the pilot are kept around in mostly their same relative roles but it's more like they take all those characters and transplant it into a new or fixed premise.
It coincided with Burn Notice pulling the whole MICHAEL HAS FOUND THE DUDES, NOW HE'S BACK IN THE CIA, OH WAIT THERE'S ANOTHER DUDE AND ANOTHER card yet again and again, so it felt extra good that this really mediocre show became a more serious and increasingly serialized spy show while Burn Notice was descending into product placement and endless reveals while dropping a lot of the voiceovers about going about spycraft and lots of terrible plots of the week with two minutes at the start and end of each episode dealing with the slowly degrading arc.
In a lot of ways, when Burn Notice had Michael "welcomed back into the CIA" just to ditch that after like five episodes to create a NEW BIGGER BADDER SECRET GUY BEHIND IT ALL AND YOUR BURNED AGAIN, Covert Affairs almost assuredly coincidentally stepped in with a show about CIA agents doing CIA stuff in deep or long term covers where things continued from episode to episode, that Burn Notice never allowed because it had to do client of the week every week and also have stuff for Sam/Fi plus now Jesse to do. Which Burn Notice never seemed to be able to get away from doing.
To bring it back around, Person of Interest didn't have that problem, much like Justified, in terms of throwing out the case of the week requirement it starts with once it finds a plot to dig into. Covert Affairs made that same change, it just took longer.
