It might sound contradictory, but I think Saints Row: the Third is a lot of fun and you shouldn't play it. SR3 was generally well received at the time, but since Saints Row IV came out there's just no reason to go back to it.
The gameplay is varied enough to keep your interest, it lasts long enough to feel satisfying and not so long that you get bored, and it remains challenging while giving you access to powerful weapons. In gameplay terms, it's basically a more polished Saints Row 2. But on the other hand, SR4 is a more polished SR3, and SR4's story doesn't have the huge, glaring issues that SR3s has – character and pacing, for example. A lot of these issues were apparently down to executive meddling – THQ cut some parts of the story without giving the developers time to rewrite – so the plot is hard to follow at times and characterisation is inconsistent.
But a bad plot is not a big deal in a video game. Its underlying message is another matter. In SR2 the protagonist was a bad person. They¹ did bad things, but we understood that this was not a character we were supposed to identify with. If that tone had continued, it might have been OK (narratively) for them to force trafficked women into prostitution (yep, that happens), because that is something that a bad person would do.
But this protagonist is not really the same character. SR2's protagonist was a vindictive, selfish, and didn't care who they hurt. SR3's protagonist (despite ostensibly being the same person) is a “puckish rogue” motivated by a desire to protect their friends; the power and value of friendship is a major theme. It's absolutely not OK for this person to do unambiguously bad things (like sex trafficking, seriously, oh my god).
The game wants to have a protagonist who's a good guy we can identify with, but also to have that protagonist do terrible things, and the result is a narrative that seems to advocate murder, exploitation and abuse. Whereas the first two games in the series showed that these terrible acts brought nothing but destruction and misery, and the later ones were wacky, over-the-top stories about friendship and courage, SR3 doesn't know which way to go.
So I don't think you should play it. SR2 or SR4 are more fun anyway, depending on which parts of SR3 you like best, and those games don't teach you that being bad is actually good.
¹ The gender of the Saints Row series' unnamed protagonist is selectable by the player, so I have chosen to use the gender-neutral pronoun “they” in this review.