1) How it started
We really have to thank Howard and especially Raj, for making the effort.
2) You've Got a Friend in Me
Sheldon and Amy started to develop a very sweet friendship, spending the next four months communicating with each other on a daily basis since they first met. I am sure they found each other fascinating since they are more similar to each other than anyone else. They know how each other thinks, they have logic that is very obvious to both of them, and only to both of them (see how they play Counterfactuals)
Sheldon's new friend Amy
Sheldon's first date
Their Counterfactuals Game
3) The first 'break up'
Of course, in any healthy relationship, we argue, we disagree and we fight. For Sheldon and Amy, their friendship 'outlived its utility' when Amy belittled Sheldon's occupation as a theoretical physicist. Sheldon terminated their friendship and adopted 25 cats as a substitute for Amy. Of course, this problem was easily solved by Mrs Cooper, using reverse psychology to make Sheldon offer to reconcile with Amy.
4) Amy held Sheldon's hand
Then, their friendship started to evolve. We see Sheldon's first sign of jealousy.
Sheldon preferred to assume that Amy has contracted an alien parasite than the reluctant conclusion that Amy is sexually aroused by Zack. Of course, to our relief, Amy chose Sheldon's intellect over Zack's looks, and even held Sheldon's hand. I like the idea how Sheldon did hold her hand back and uncharacteristically did not pull his hand away and he let his hand be held.
5) Fun together, as friends
They have lots of fun together as friends. They conducted social experiments (Look at you, getting me to engage in the social sciences. You're a vixen, Amy Farrah Fowler.), they danced, they kissed. Of course, Sheldon may not want to cross the line of being just friend so they agreed to treat their relationship like a "crashed computer" and restored it to the last point it worked for both.
Amy getting Sheldon to engage in the social sciences
Amy and Sheldon dancing
Amy kisses Sheldon
6) Sheldon got jealous
However, no matter how Sheldon wishes to deny, still, we do know that he has feelings for Amy. When Leonard told Sheldon that he had a great time with Amy, Sheldon karate hit him on the shoulder and angrily retorts "She is not for you. Not for you!"
She Is Not For You
7) They cuddled
When Amy got really upset, and not returning any of Sheldon's various digital attempts to contact her (I tried email, video chat, tweeting her, posting on her Facebook wall, texting her, nothing), Sheldon came into her rescue. Amy made clear what Sheldon could do for her: “At this moment, I find myself craving human intimacy and physical contact.” So, even though Sheldon is uncomfortable with human physical contact and has germophobia, they cuddled......
Sheldon And Amy Cuddle
8) Sheldon wanted to win her back
When Stuart asks Amy out on a date, Sheldon seems to not mind though it becomes obvious later on. He friended Stuart on Facebook to find out how they date, he tried to make Amy jealous by trying to ask Penny out for a date. Even though Sheldon still insists that Amy is a girl who is his friend but not his girlfriend, he wanted to know what are the ways to get her back.