The Companions are usually the first faction people stumble into, mostly because Jorrvaskr is right there in the middle of Whiterun and a giant's stomping around Pelagia Farm when you arrive. So here's the full arc, start to finish, plus the one decision people actually agonize over: keep the wolf blood or burn it out.
Getting in and the early jobs
You join by talking to Kodlak Whitemane, the Harbinger, after Aela and the others help you put down that giant outside the city. Kodlak sends you to Vilkas for a sparring test in the Jorrvaskr yard, then you fetch his sword from Eorlund Gray-Mane at the Skyforge. That's it. You're a whelp.
The early stretch is radiant filler: retrieve-an-item, clear-a-camp, shield-a-civilian jobs handed out by Aela, Skjor, Farkas, and Vilkas. None of it is mandatory in any deep sense, but "Proving Honor" is the real first story beat. Farkas goes with you to Dustman's Cairn to recover a fragment of Wuuthrad, Ysgramor's legendary axe. Mid-dungeon you get ambushed by the Silver Hand, and Farkas casually turns into a werewolf and tears them apart in front of you. So now you know what the Circle, the inner five, actually are.
The Silver Hand and becoming a werewolf
"The Silver Hand" sees Skjor and Aela induct you into the Circle. You drink Aela's blood from the font in the Underforge and take the beast blood yourself, then go raid Gallows Rock. Skjor dies there. If you didn't think the Silver Hand were a real threat, that's the game correcting you. The Silver Hand are werewolf hunters, basically the Companions' shadow, and "Blood's Honor" / "Purity of Revenge" are the revenge run after they hit Jorrvaskr and kill Kodlak while you're away chasing Wuuthrad fragments at Driftshade Refuge.
The quiet tragedy under all of it: Kodlak never wanted the wolf blood. He believed it barred him from Sovngarde, sending Harbingers instead to Hircine's Hunting Grounds for eternity. He'd been looking for a cure when he died.
Glory of the Dead
The finale. You carry the now-complete Wuuthrad to Ysgramor's Tomb with the surviving Circle, fight through draugr, and enter the Tomb proper where you can cure Kodlak's spirit. You toss a witch's head into the flames to free him. He goes to Sovngarde, you're named Harbinger, and that's the questline done.
Worth knowing: the Harbinger isn't a chief. Kodlak's whole point is the Companions don't take orders, they take counsel. So "leading" them is mostly ceremonial. The radiant jobs keep coming afterward if you want them.
Keep it or cure it
Now the actual choice. After Glory of the Dead you can cure your OWN lycanthropy by collecting another witch's head and burning it at the same flame (the "totems of Hircine" side content lets you tweak the beast form too, if you keep it).
Keep it and you get the werewolf transform once per day: big melee burst, fast movement, feeding heals you, and you're immune to disease. No vampirism risk, which matters. The downsides: you don't get the Rested/Lover's-Comfort sleep XP bonus while you carry the blood, and you can't absorb dragon souls in beast form (minor, since you'd transform back).
Cure it and you re-enable the resting bonuses, you can become a vampire later if you want Dawnguard's Vampire Lord, and thematically you've honored Kodlak's wish. Most of my pure-stealth-archer characters cure it because the daily-cooldown brawl form just doesn't fit how they play. My two-handed Nord brutes keep it forever.
Honestly the "is it good" answer is build-dependent and a bit underwhelming either way by the time you're high level. The transform doesn't scale well, which is its own complaint.
The real gripe
For a flagship faction with this much lore weight. Ysgramor, the Five Hundred Companions, Wuuthrad, Hircine, a Harbinger you actually grow attached to. The whole thing is maybe seven story quests and you can blitz it in an afternoon. You're handed the top job after one dungeon crawl as Harbinger. There's no politics, no consequence, no faction-vs-faction tension like you'd half expect.
Does the Companions arc end too abruptly for what it's setting up. And would you have traded the radiant filler for two or three more real story quests about actually being Harbinger?