Dork Age: The "guest-writer" era that came from #22 all the way through #44. Although there were some well-received stories, for the most part the majority of them are considered a huge step down in quality, often portraying the book as Lighter and Softer and overly comedic (as noted by Bizarro Episode up above) rather than the dark action-adventure tone. Eventually Eastman and Laird themselves stated that only two issues from that era (#27-28) were canon.
Growing the Beard: As good as the comic and the franchise itself would later get, the original first issue which started it all was... underwhelming. The turtles all had the same one-note personality, the art was a bit shabby, and the Shredder was killed off in the first issue. Once that initial parody one-shot was turned into an ongoing series though, the creators realized that they needed to up their game. They weren't always successful at first though; with the Shredder already dead, where else was there to go? They tried throwing everything at the wall and seeing what stuck, but the comic still didn't have much of a voice other than just being kind of weird. On the other hand, they started developing the Turtles' personalities and gradually turned them into the characters we know and love today. By Issue 10, the Turtles had become fully fleshed out characters, and on top of that, they both brought back the Shredder and turned Casey Jones into a supporting character. From this point on, while the series would still delve into the weird and the strange, it had settled on being more of a serious action-adventure comic that occasionally ventured into Fantasy Kitchen Sink, and continued to develop the Turtles and their supporting cast further while actually letting the Shredder stick around long enough to be a proper threat. The comic finally earned the success it had gained, and it shows in the writing and the artwork, all of which was a major step up from that first issue. And that tenth issue was the moment where everything finally came together into both a cohesive and awesome whole.
Harsher in Hindsight: In Baxter's first appearance, he threatens to use his mousers to tear apart the structure of one of the buildings of the World Trade Center and make it collapse upon itself.
Moment of Awesome: The Turtles' first battle against the Foot Clan, humiliating the Shredder by easily killing his best troops, besting him in combat, and foiling his Taking You with Me attempt.
Also, that first comic in general, for far outperforming Eastman and Laird's idea of how much it would sell, and starting one of the biggest franchises of all time.
Signature Scene: Either the cover of the first issue, or the turtles leaping to attack the Purple Dragons in the two-page splash of the same issue.
Another one from Issue #1 is the battle against the Foot and the Shredder on the rooftop. Most incarnations since have had a battle against the Shredder that took place on a rooftop.
Seinfeld Is Unfunny: The amount of violence, especially in the early issues, were pretty shocking for the time, but considering how tame they are compared to several mainstream comics nowadays, newer readers may not be able to see why they made such a big deal.
What Do You Mean, It's Not for Kids?: No thanks to the 80s cartoon, which is far more well known. This comic could be mistaken for another child-friendly comic when it's not bloated with violence and gore.