This is such a mammoth topic, I'll try to keep it at the very highest level of detail or I'll probably go insane trying to describe the near-infinite ways in which this show has screwed up.
1.
Not building the story around core character arcs. Not even
recognizing the core character arcs that would serve as great things to build a story around. Peter is Luke Skywalker. Nathan is the corrupt soul trying to get clean. Matt is the nice guy struggling with very not-nice powers. Mohinder is the noble scientist who is undone by his scientific curiosity. Noah is the protective father undone by his overwhelming love for his daughter.
Note that those last two don't require superpowers. This is a story about
people, not superheroes, and there is nothing whatsoever wrong with Mohinder staying "mortal."
The writers needed to assess their characters after S1 and latch onto the ones with powerful character arcs that could propel a story. The other characters are killed off or relagated to secondary status. Nikki - no idea what she's good for. Kill. Claire - secondary to Noah's story since she's not interesting in her own right. Keep, as an adjunct to other characters' stories (not just Noah; she also fits well with Peter's and Nathan's stories.) Hiro - redundant with Peter's role but without the intrinsic connections to the other characters. Kill.
A few characters could have been added, such as Micah. The precocious-but-wounded orphan struggling to survive and grow up in adverse circumstances is a very durable arc and would fit well into the overall mix. But that gives us six major characters to focus on, and that is more than enough for any show. Concentrate on those, and be very careful with adding more to the mix. There is more than enough on the writers' plate.
2.
No overall story arc. The individual character arcs should be corralled together to create a single arc along a theme. The obvious overarching theme is one that dates back to ancient Greek myth. A mortal is given the power of the gods, but being mortal, cannot wisely control the power and is destroyed by it (unless he manages to divest himself from the power in time).
That's the
Heroes theme. These characters have one of three fates: destruction; permanent return to mortal status by their powers being taken away; or (maybe in one instance, and only after the greatest trials) transmogrification through wisdom into a godlike character capable of coming to terms with their power.
The character arcs will lead to one of these destinations and that's the whole story of the whole show over all the seasons. Simple, but elemental enough to provide years' worth of stories. Why did they have to mess it all up with irrelevancies and nonsense?
3.
Sylar - argh. Kring and the audience were so dazzled by Zachary Quinto that nobody stopped to think about Sylar. He's a no-win character. If he's bad, he's repetitive and boring. If he's good, he's wimpy and boring.
I think this has got to be the biggest gulf I've ever seen between the caliber of the writing for a character (botched from the start - his motive was weak and whiny and needed to be rethought before the cameras ever started rolling) and the acting (so wonderful that, terribly, the botched character was neither killed nor un-botched).
I'm not sure if there was ever a way to salvage Sylar. Maybe just postpone the inevitable via an arc that has him evil in S1; play around with redemption in S2; and fall into the pit of even greater evil in S3, with his permanent demise at the end of the season. He was never a character concept that had a long life in him.
From what I understand, the original idea for the show had each volume be about a different set of characters,
If there is therefore to be no character arcs, that just makes it all the more crucial that the series have an overarching theme. Here is the tragedy of mortals aping the powers of gods in S1...and S2...and in S3, it's the same shit all over again! These idiots never learn. I guess that could have worked. But when did this show ever have a plot arc and coherent theme? What is this show
about?!?