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Ep. 1:  
  • Applejack is Twilight’s first friend.
     
    Ep. 2:  
  • Regardless of having just met her, Applejack defends Twilight from Rainbow’s accusation on her being the enemy.  
  • AJ insist that the gang follow Twilight into the Everfree forest for protection.  
  • AJ scrambles to save Twilight’s life, even as there was no one to save the farmpony should she fall off too.  
  • Twilight trusts Applejack with her life after just meeting her.  
  • A.J was the first mentioned of the ponies that moved Twilight greatly.
     
    Ep. 3:  
  • Twilight is hanging with A.J at the beginning of the episode, helping her long before the Applebuck season incident.  
  • In Twi’s hypothetical situation, A.J is not among those who didn’t get a ticket to the gala.  
  • Applejack’s “favor” was probably the closest to what Twilight really needed at that time.  
  • Applejack was first to give up her ticket, even as some may argue, that she had the most urgent reason to go.
     
    Ep. 4:  
  • At the award ceremony, Twilight, and not the mayor, is hosting the event for Applejack.  
  • Note the stack of papers: Twilight apparently has a lot of good things to say on AJ.  
  • Everyone notices something’s off, but only Twilight visits AJ to inquire about it.  
  • A.J sense? No one fingers (or hooves) AJ as the reason for the problems in town, yet Twilight somehow knew AJ needed her attention.
     
    Ep. 7:  
  • A.J saves Twilight’s life again  
  • Twilight gives her an affectionate nuzzle in thanks. This sort of sentiment is not done at any other time after a recuse (by any of the mane 6, on each other) in the future.
     
    Ep. 8:  
  • Along with Rarity, A.J tries to make Twilight’s sleepover a success despite her feelings on being around another pony.
     
    Ep. 11:  
  • Applejack’s rebuke actually affects Twilight enough to make her cry (first time too).  
  • The two share a sister-hoove bump for a job well done (there’s no other physical exchange done with Twilight and any other of her friends during that montage)
     
    Ep. 12  
  • After the lesson is learned at Diamond Tiara’s party, Twilight and Applejack toast their glasses together and remain close while drinking. Twilight does not use her magic to lift the cup.
     
    Ep. 13:  
  • A.J invites Twilight to oversee the competition. When Rainbow and her laughs at Twilight entering the race, A.J tries harder to be more supportive and hold it in.  
  • Unlike Rainbow, A.J takes Twilight’s word on Dash and the tripping accident.  
  • Twilight tells AJ to “be careful” during the race, but doesn’t extend the same word of caution to Dash.
     
    Ep. 14:  
  • A.J and Twilight are hanging out together again at the beginning of the episode.
     
    Ep. 16:  
  • A.J helps Twilight out with the library but doesn’t voluntary to have a spell tested on her to help Rainbow.
     
    Ep. 19:  
  • Twilight is the one who tries to save A.J when she’s being pulled into the Diamond Dog hole.
     
    Ep. 21:  
  • Twilight stays on AJ’s side of things to help with the convincing. Rainbow and Spike are left alone to do the convincing on the buffalo’s side.
     
    S2 Ep. 1:  
  • Twilight is quick to find Applejack first in the maze after everyone separates.  
  • Only Twilight seems to know that AJ’s lying here, even on minor things
     
    S2 Ep. 2:  
  • When Twilight asks the others if they cared, Liar-AJ says “no”.  
  • Going out to the edge of town, Twilight goes out of her way to “fight” for A.J first and win her back.  
  • In AJ’s memory flashback, there are more images of her and Twi then with any other one pony.  
  • A.J is the only one to ask Twilight for forgiveness after turning back.
     
    S3 Ep. 3:  
  • AJ does more to try and console Twilight than the rest of the group when she tells them about her issue.
     
    S2 Ep.4:  
  • Twilight tells Luna that Applejack is “one of the most likable ponies around”.  
  • Even as she’s afraid, Applejack is quickest to help Twilight with Luna.
     
    S2 Ep.6:  
  • A.J chooses to go to Twilight for help on her sister illness first, rather than a doctor.
     
    S2 Ep.7-8:  
  • The two of them comment on Rainbow’s antics privately (thought up the Mare Do plan together?).
     
    S2 Ep.9:  
  • Twilight tells Rarity that something “so simple” and “practical” is “perfect” for her, than states that she loves it. These are traits attributed to Applejack.
     
    S2 Ep.10:  
  • Of her friends, Twilight goes to A.J, and not Fluttershy, for help on getting back Spike (an out of control dragon), showing once again that the two rely on each other quite a bit.  
  • The two of them wind up tied awkwardly to a tree together. For some reason, Rainbow Dash suggests that they did it to themselves.
     
    S2 Ep.11:  
  • In the play, during the cave territory marking scene, the two accidentally bump hooves in an awkward moment.  
  • Applejack and Twilight’s characters are two of the few that truly get along.
     
    S2 Ep.14:  
  • Twilight ends up furthest out on the platform, running to catch A.J’s train as it departs.  
  • Twilight’s first to leave the assembly line to question AJ.  
  • Twilight is first to speak up to cheer a dishearten A.J.  
  • Twilight seemly forgets about Rarity and Pinkie afterwards when they have AJ, even though she was the one who worried about the two being left behind in the first place.
     
    S2 Ep.15:  
  • Somehow, Twilight is always in line ahead of most of the others (and all of her friends, minus Pinkie), for more than just one day for the cider line (does this mean she gets up earlier and loves the drink more than Rainbow Dash?).  
  • Applejack confines in Twilight about her true worries and the latter checks up on her through the episode.  
  • Twilight is only one who sticks by AJ throughout the whole ordeal.  
  • Twilight apparently remembers AJ’s talk about her being “part of the family” as she cooks up a plan using the “honorary family member” phrase.  
  • Even being at different ends of the cider making assembly line, both collapse next to each other and share a moment.
     
    S2 Ep.25:  
  • Applejack is the one who congratulates Twilight on her brother’s wedding.  
  • AJ comes up with the P.F.F lingo (pony friends forever).  
  • On the train, Applejack checks up on Twilight, trying to comfort her about what’s been upsetting her and pushes her to talk to Shining Armor.  
  • Twilight first hangs with A.J during the wedding preparation.
     
    S2 Ep. 26:  
  • Applejack trusts her back to Twilight (or who she thought was her)  
  • Twilight saves A.J during the fight, and despite what had happened, Twilight takings just AJ’s WORD that she’s not a changeling.  
  • The only one who apologizes to Twilight for not believing in her is Applejack.
     
    S3 Ep. 1:  
  • AJ embraces Twilight after the “wasn’t prepared for it” song.  
  • AJ was the only one who came through for Twilight with information. She reminded everyone they were here for Twi and even took over for the purple unicorn when she was overcome by the library.
     
    S3 Ep. 5:  
  • Applejack tries, unsuccessful, to stand up to Trixie to bring Twilight home.  
  • Twilight chooses Applejack to transform into a stallion.
     
    S3 Ep. 7:  
  • AJ was on her way to see Twilight to observe her studies when she came across Pinkie.
     
    S3 Ep. 8  
  • Applejack channels Twilight from Lesson Zero.
     
    S3 Ep. 9:  
  • AJ knew exactly how to get Twilight’s attention.  
  • Frighten she scared her, AJ apologizes to Twilight over moving one of the ink pots.
     
    S3 Ep. 13:  
  • Only A.J recognized Twilight during the transformation  
  • AJ is first to walks up to Twi after she changes.  
  • Applejack is the closest of the mane 6 to the center of the crowning ceremony.  
  • Twilight actually makes A.J cry, for a good reason.
     
    These two are close.

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Background Pony #CD12
I didn’t take the time for these type of observations like you did OP, but I like these 2 small moments too.  
fullfull  
It seems AJ and Twi like to push each other around a bit :p