Shoot relationship timeline
Following their first meeting, Shaw begins hunting down Root out of anger, but there is also a definite element of fascination. Root managed to trick her completely, which is not easy to do given what we've seen of Shaw's cold-reading skills, and shares her enjoyment of violence.
Later, Root gets the drop on her again by showing up in her bedroom to kidnap her. Shaw is furious and only agrees to work together once Root proves she's getting orders from the Machine. Throughout this mission, she witnesses Root's loyalty, considerable skills and the promised violence (as well as other things in the CIA safehouse) that prove their compatibility.
While Root is in the Faraday Cage, Shaw consistently advocates for letting her help them. She still doesn't trust Root, but understands her and knows she will do what she's offering. Sure enough, Root is let out of the cage and saves Reese, which is when she earns Shaw's respect and trust.
Shaw now views her as part of the team, demonstrated by the way she easily begins working with Root when she rescues them from Control and is reluctant to leave her with Hersh, then goes to look for her at the end of the episode.
Shaw doesn't see Root again until she returns to New York in 3x17, at which point she displays concern for Root's welfare when she checks the dressings on her injuries. This could be because it weighed on her mind that the last she saw of Root was evidence she had been tortured, only to learn she had left the country with nothing but a brief message to Finch.
I headcanon that they began sleeping together again after this episode and it cements them as being on genuinely friendly terms. This is shown by Shaw enjoying the opportunity to do a relevant number with Root (including lots of violence), showing an interest in her life and suggesting a steak joint, implying that she wants to spend more time together, in 3x20.
I would also suggest it's around this point Shaw starts to realize Root has feelings for her, as Root's flirting progresses to adoring stares (3x17, 3x20), taking opportunities to touch her (3x22) and compliments ("She knows you can handle yourself." 3x22), in a way that seems genuine, rather than just trying to annoy her.
They continue as friends-with benefits, bantering and looking out for each other, until the turning point in 3x23. Upon learning Root is going into a Decima fortress alone and there's a good chance she'll be killed, Shaw stops in her tracks and decides to help her immediately. This is the moment she realizes Root means a lot to her, possibly in a different way than she's used to caring about people, and she can't stand the thought of her being in danger.
Although Shaw doesn't acknowledge this revelation in her interactions with Root, she clearly finds it difficult to walk away from her when they have to go their separate ways as Samaritan comes online.
In season 4, they don't get to spend much time together due to keeping up their cover identities. But Shaw still pays attention to and cares about Root, as she immediately knows something is wrong in 4x05.
In 4x07 Shaw meets Tomas, and is both attracted to him and intrigued by the life he's offering. But when the time comes to choose who she wants to be with, she picks Root. I would say this is the moment Shaw realizes she loves her, because although she sees the appeal of a glamorous life of crime abroad, what she really wants is to keep what she has with Root. She then goes to find Root, cites her reason for staying as "I guess there are things I care about here." and finds an excuse for them to spend time together. This is as close as Shaw gets to declaring her feelings.
Neither of them take the leap to change anything about their relationship, but their following interactions carry an undercurrent of intensity because they know what they mean to each other. Such as Root telling Shaw that "while you may not be scared about what could happen to you the next time, other people are. People who care for you." (4x09) and Shaw shoving grenades into her bag while saying, "She's going to meet an all-seeing, all-evil god by herself. That doesn't sound fine to me. That sounds like someone who needs backup." (4x10)
In 4x11, before sacrificing herself to save the team, Shaw kisses Root. Knowing it is her last opportunity to do so, she allows herself to embrace how much this brilliant, frustrating woman means to her.
Root is already a fan of Shaw before their first meeting, having read her file and been impressed by her considerable skills. But, since misanthropy has been the driving force of her life, she is likely also intrigued by the idea of a heroic sociopath. And sure enough, once they meet it's a case of intense thirst at first sight, only furthered by Shaw's declaration that she "kind of enjoy[s] this sort of thing."
During their first mission together, Root clearly enjoys messing with Shaw ("Safety first." 3x06) and is thrilled to learn that she lives up to the reputation in her file. But Root also recognizes that Shaw actually listens to everything she says and acts accordingly. After a lifetime of being dismissed as crazy or a liar, that's significant.
In 3x12, when Root learns that Shaw came to look for her, she is genuinely touched and I would pinpoint this as the moment she realizes she has feelings for Shaw. She's been alone for nearly all her life, and now this incredible woman cares about her enough to come back.
When she returns to New York, her flirting is as blatant as ever but seems softer ("I love it when you play doctor." 3x17), as if she means it with more sincerity but also has something to lose. As mentioned above, this is around the time they start sleeping together again.
Root continues crushing on Shaw, but doesn't do anything to change the status quo. Until 3x23 when Shaw comes to save her at the Decima fortress, which is the first hint that Shaw likes her back. Root fully expected to die on that mission and sounds unusually serious when she says, "Admit it, you were worried about me." But when Shaw's reply doesn't reveal anything, she continues with their usual banter.
The next development happens in 4x05 when Root, again believing she's going to die, asks Finch to give Shaw a message. While unwilling to approach the subject with Shaw directly in case it violates her boundaries, she views her feelings as significant enough that she wants Shaw to know if the worst happens.
In 4x07 when Shaw chooses to stay with Root, she is clearly thrilled. Following Shaw's confession she teasingly asks, "and is that why you came to see me?", but the answer doesn't matter. She knows how Shaw feels, her actions say everything.
As stated above, neither take the plunge to change their relationship after this. When Shaw sacrifices herself, Root is devastated and willing to do anything to find her.
After they are reunited, they have both been holding onto their love for those nine months. Root never stopped looking for Shaw or believed she would betray them, Shaw viewed Root as her safe place. So I headcanon that shortly after their meeting in the park, they have a conversation about their relationship and are officially 'together'. This is shown in 5x10 by their easy, flirtatious banter ("You want to stick around, give 'em a proper welcome?" "Thought you'd never ask.") and tender, vulnerable conversation while holding hands ("This might be the first time I feel like I belong.").
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