The rumor that Vriska made up about Terezi is what is needed for her friends to start judging her. Each of her friends have a reason to not believe this rumor, but teenagers can forget things easily. It took time though, more time then Vriska wanted it to take. It's hard to say exactly when Terezi's friends fell into Vriska's clutches, but we know that the first few people were Tavros, Kanaya, and Equius.
It's unknown how, but I am giving you my ideas to how each of them started to believe Vriska over Terezi. Starting with Tavros: As we know, Tavros doesn't have a lot of self-esteem, and he has a crush on Vriska, so it was fairly easy for Vriska to convince Tavros to believe her and not Terezi. Tavros probably just went along with Vriska from the start, but we don't know that for sure.
Kanaya was a bit harder. Kanaya had some "I owe you"s to Terezi for help, but Vriska and Kanaya have been friends since before school. They used to live right next to each other, so Vriska knows everything about Kanaya, which made it easier for Vriska to "convince" Kanaya to leave Terezi. Before Kanaya did though, she gave Terezi $50 dollars for the "I owe you"s. Terezi didn't quite understand in till much later.
Equius was an interesting case. He happened to have a gigantic crush on this one girl in school. Vriska knew this, and with some manipulating and persuasion, she was able to get Equius and the girl dating, which took any idea of Terezi from him.
Three friends down, six friends left. With the little time the Vriska had left, she had to work quickly, if she wanted Terezi's summer to be ruined. Of course, this is assuming that Terezi wants the same summer as Vriska wants. Terezi was actually just spending a lazy summer, spending half of the time by herself, the other half with Sollux. Sollux is planning on working during the summer, which is a good idea.
Back to the story, Vriska targets Feferi and Eridan. Since the death of Feferi's sister, Feferi hasn't been like herself. At first, she closed herself off from everyone except Eridan (who was the only person who didn't let her bottle everything up), but slowly, she was opening up again. Eridan was a bit help to her, and rumors started to spread around that they were in a relationship. Feferi always denies it, but Vriska knows what she could do to make Feferi and Eridan turn against Terezi.
Vriska teases them at the beginning, but when the teasing got out of hand, that's when Vriska tells them that Terezi is the one that persists that they are together. Feferi tried to ask Terezi about it, but it failed because of Vriska. With that, Terezi began to think that Vriska is trying to do something.
Five friends, four friends left. Nepeta was already drifting away from Terezi, because her best friend Equius was dating a girl he liked, and Nepeta always goes with Equius. Vriska thought about making the relationship bad between Nepeta and Terezi, but she quickly lost thought with what to do with Gamzee.
Terezi and Gamzee have a strange relationship. Sometimes they would hang out and just drink Faygo together. Vriska thought that she could do something very complicated, but it's actually very simple. Just make Terezi say no to hanging out with Gamzee. Terezi has never said no, so to make her say no will have a big impact. Actually, Vriska failed with this one. Terezi and Gamzee didn't hang out before the summer as Vriska tries to do this.
Vriska also tried with Karkat and Sollux. She came up with many ideas, and tried many of them. Each failed like a burning pirate ship. Sollux would always dismiss her much before she could even get past the second step of the plan, and Karkat, well, was just his usual self.
Because of this, Vriska had to find out about the relationships that Terezi has with each of those three boys. With Gamzee, they are just pleasant friends that just mellow out with each other when Terezi needs it most. With Karkat, he shows loyalty to Terezi for the promise he in trusted her with (That Vriska never finds out about) and for all the times that Terezi has helped him out of trouble. With Sollux, he owes her for when he lost Aradia, and we know why he also wouldn't leave her.