Steve:
Steve asked you how you wanted to meet the Avengers and you thought that it might be nice to host a dinner for them. Steve invited them and their dates all over to his place and the two of you cooked dinner together. Steve was nervous about Tony saying anything embarrassing in front of you, but honestly you thought he was hilarious and when he occasionally did make Steve nervous and caused him to blush you just thought that Steve was even cuter.
Bucky:
Bucky introduced you to Steve first, by having the two of you go out for drinks with Steve and his girlfriend. Then after hanging out with Steve a few times he introduced you to the rest of the Avengers at a Stark party that Tony was throwing to honor some achievement that he had made. You were nervous at first, but having Bucky's strong arm secured around your torso made you feel more comfortable. Also already knowing Steve and his girlfriend made meeting people less intimidating.
Pietro:
You had met Wanda at the same time that you had met Pietro, so he didn't have to introduce you to her. You had also knew Tony before meeting the twins, and of course Tony had introduced you to Bruce so you knew both of them well. As for the other Avengers, Tony invited you and Pietro to a Stark party and you met them at the Party. Not until the end though, when most of the other guests had already gone home.
Loki:
Loki has introduced you to Thor one night when he had accompanied Loki back to your apartment upon their return from Asgard. In the cases of the other Avengers, Loki did not really associate with them after everything. Which meant that you were not introduced to any of them or got know any of them in person.
Clint:
Clint asked Tony to throw a mini Stark party for you to meet the whole team. Tony wanted to go all out for the night, but he kept it small because Clint insisted it would be better for you that way, and because the purpose was for you to meet the Avengers. He was right, of course, you enjoyed yourself and you got the opportunity to get to know every member of the team. You also got to relax and party a bit, which was quite fun.
Thor:
You met Tony first because Thor asked him to keep an eye on you while he was away in Asgard and he stopped by a few times, at first you were annoyed because you felt like you were being babysat. Eventually you realized that Thor had done it because he cared, and you had already forgiven him by the time he returned. When he did return he took you to the Avengers tower to introduce you to the rest of the team members.
Tony:
The Avengers are always around Tony's place, and since you live with Tony you ran into them all of the time, and ended up meeting them mostly one at a time. The first you met was Bruce because of course Tony is basically attached at the hip to the man. After that you met each of the other Avengers at different times until eventually you knew them all pretty well.
Natasha:
Natasha invited each of the other Avengers to your place at separate times for them to each meet you. This way she wasn't drawing too much attention to your home and you wouldn't be together out in the open. You met Clint first since he was her best friend and he told you a bunch of stories about Nat from missions they had together, which had Nat red faced and you laughing. After that you met Steve who as really kind and then the others after that.
Wanda:
The Avengers were all headed out for a week of training together, at Fury's request. Wanda had asked you to drop her off, which you agreed to without hesitation. When you dropped her off everyone else was already there so Wanda introduced you to all of them. Although, you had already met Pietro when you and Wanda first started dating because he was watching out for his sister and therefore wanted to interrogate you.