As far as relationships go, the Avengers have generally stayed away from dating within the group. Black Widow and Cap kissed once, and Widow and Banner revealed their affections for each other in Age of Ultron, but other than that they've seen their fellow Avengers as brothers-in-arms (and sisters-in-arms).
However, after Civil War, that might be changing. Warning! There are many major and minor SPOILERS for Captain America: Civil War. If you have not yet seen it and you don't want it spoiled, click away now.
Even though the film is labeled as a Captain America movie, all of the Avengers were there front and center, and the different friendships and relationships were explored in-depth. Cap got to revisit his friendship with Bucky and start a new relationship with Agent 13, Black Widow and Hawkeye had a cool little friendship moment, Tony and Rhodey got to fight together like the good ol' days, but there was one friendship/relationship that got a major jump-start:
Scarlet Witch and Vision!
Though this may seem like an odd pairing between a young, enhanced, Sokovian girl and an all-powerful android with an Infinity Stone in his head, this romantic couple isn't as far-fetched as you may think. In fact, in the comic books, they actually get married!
When it was announced that Vision and Scarlet Witch would both be making their debut in Age of Ultron, many comic fans were ready for their relationship to be set up, which in a way came to fruition when Vision rescued Scarlet Witch at the end of the Sokovia battle.
While their romance might have been teased in Age of Ultron, it definitely progressed in Civil War. Almost every interaction between the two heroes screamed romantic potential. During the conversation about the accords, Wanda asked out loud if they would come for her, which Vision instantly replied with: "We would protect you."
Later on in the film, Wanda and Vision have a very personal scene where Vision makes paprikash to help lift Wanda's spirits. It is then that Vision starts up a conversation, which be prefaces by saying: "Nobody dislikes you Wanda." Throughout the entire conversation he tries to make her feel better, explaining that she's only feared because people don't understand her.
Unfortunately, seeing as the two of them were on opposite sites of the Civil War, they had a little bit of conflict throughout the movie. You could still see the pain behind the conflict, like Wanda's sadness when she manipulates Vision so she can escape the Avengers compound with Hawkeye.
The biggest nod to their eventual romance was at the end of the airport battle. As Vision was cradling Wanda after the battle wore her out, Rhodey asked Vision for help with Falcon. Vision shot a blast from his mind gem at Falcon, but Falcon dodged, and the blast took down Rhodey instead.
When Tony later asked how that happened, Vision explained that he was distracted. Tony said that he didn't think that was possible, and Vision replied: "Neither did I." He was distracted because he was with Wanda, and he didn't think that as an android, he would be able to fall in love. But it's very clear that he loves Wanda, and that he's just starting to learn human affection to the best of his abilities.
While this may seem like an interesting idea to start up a relationship between two Avengers, what they are actually doing is setting up a tragic romance because sadly,
Vision Is Going To Die!
As you may have picked up from Guardians of the Galaxy, one of the largest running themes of the MCU films is the Infinity Stones. So far every moment in the MCU has been setting up Avengers: Infinity War, where Thanos (the purple-faced alien at the end of The Avengers) will acquire all six Stones and all of the heroes will have to fight him in one final major battle.
Unfortunately for Vision, the Mind Stone is the very yellow gem used to power him, so if Thanos is to recover all of the gems, that means he must destroy Vision in the process.
With the Avengers' currently broken and scattered state, it might seem like Wanda and Vision won't have too much time to progress their relationship. That being said, Infinity War is rapidly approaching, and it would make sense for their relationship to intensify before Vision meets his ultimate fate. I guess we'll just have to see where the story takes them.