The only people who translated from Saints Row 2 are The Boss (you), Shaundi, Pierce, and Johnny. From that time, the Third Street Saints have become humongous celebrities and have their own clothing and food chains. The game begins with you doing a bank heist in Stilwater (SR 1 & 2's setting) that goes awry because the bank is heavily protected by a group called The Syndicate -- which is a syndicate of the main gangs in Steelport (SR3's setting). The Syndicate, lead by a kingpin of The Morning Stars gang named Phillipe Loren, takes you out of prison and offers to have the Third Street Saints join The Syndicate for their lives, if they give up most of their wealth. The Boss obviously says no, Johnny dies in their escape (in the first ten minutes of the game!). Shaundi, The Boss, and Pierce crash land in Steelport with a vendetta to destroy The Morning Star, get a strong foothold in Steelport, and kill Phillipe.
A little bit later, you kill Philip of The Morning Stars, but the Syndicate is immediately taken control by Killbane, a former wrestler and current kingpin of a wrestling-themed gang The Luchadores. On the mission to kill Phillipe, you meet a brute named Olaf, a Russian heavy and certified genius, who joins the Saints as a means to take revenge on Phillipe. Afterward, at a memorial to Johnny's death, The Luchadores crash the party and the Saints (joined by Olaf) take objective to destroy them.
They first rescue Kinzie Kensington, an ex-FBI hacker, who was taken by another gang of The Syndicate, a goth techno gang called The Deckers. Then they rescue Zimos, a pimp who only speaks in autotune, who had been used as a sexual human pony in a nearby brothel owned by the DeWynter sisters (DeWynter sisters are a part of The Syndicate, but have no gangs). (look up the mission, shit is hilarious). Then they help an ex-wrestler Angel, who was shamefully defeated by Killbane and wants revenge. Viola DeWynter, who runs the prostitution in Steelport and is partly responsible for Zimos's time as a human pony, defects from The Syndicate and joins The Saints after Killbane murders her sister, Kiki.
The same time, a military group named STAG is deployed to Steelport because of the rising gang violence, lead by a grizzled veteran named Cyrus Temple.
Using Kinzie's hacking skills, The Boss eventually breaks apart The Deckers (lead by Matt Miller). Killbane is defeated later, in an event called Murderbrawl XXI, where Angel and The Boss tagteam him and defeat him in the ring of honor. (Look that mission up too, super fucking fun.) However, Killbane refuses to lose control of The Luchadores, and STAG is still prevalent in the city.
The game ends with two choices hitting The Boss, as the remains of the Luchadores and STAG go on an all out war on Steelport: First would be to murder Killbane (and later Cyrus on a giant fucking fortress) in a long act of destructive revenge, and declare Steelport independent of America. Second would be to murder STAG's second in command, Kia, and save Shaundi, Viola, and Mayor Burt Reynolds (you fucking read that right, Burt Reynolds), where you would then order STAG to leave Steelport because STAG nearly blew up a national monument to kill Shaundi, Burt Reynolds, and Viola.
In Saints Row 4, the 'save Shaundi, Viola, and Reynolds' option is canon.
