I was never really into anime growing up. The extent of it was Dragonball Z, and that was a longgggggggggggg time ago. It's only until recently that I started picking it up again, and that mainly started with my consumption of Cowboy Bebop.
Cowboy Bebop is long considered a "gateway" anime, which I agree and disagree with. I agree because it's pretty good for Western audiences with it's faster pace, crisp action sequences, and clever dialogue. The fact that the dubbed voice acting is superb (Steven Blum and no one else is Spike) doesn't hurt it either. It's probably as close to a "Western" cartoon that happens to be an anime. And the fact that a lot of its influences are from Western pop culture (the use of Jazz and it's heavy references to Blade Runner), it makes it perfect for people like me who haven't watched anime that much.
However, for the same reasons, I disagree with the notion that it's a gateway anime because it's style and storytelling is different from common genre staples in anime. Instead of episodes being long and drawn own, each is episodic and often wraps up nicely in the end. Things move quick in Cowboy Bebop, with the story jumping from scene to scene instead of the slooooooowwwww pace I've seen in other animes. And, most importantly, the plot moves on its own. I've noticed after watching other animes (like Death Note....ugh) there's A LOT of exposition. Sometimes I'm sitting there for minutes while the main character describes every meticulous detail of his plot. Cowboy Bebop rarely does this, which is nice.
When thinking of my favorite episodes, Asteroid Blues (the first one in the series), didn't even come across my mind. However, as I watched it over again, it made me realize that it's the perfect introduction to the world of the Bebop crew. It's hard to understand things at first. There's little set up to the future world Cowboy Bebop has created and little explanation. Instead, you simply jump right into the action, picking up on things as you go.
The episode centers on a drug called red eye, a hyper steroid of some sort that makes Bane's venom look like child's play. After a demonstration of its power, we follow Spike and Jet on their quest to get the bad guy and collect the bounty. It's pretty straightforward plot wise, but it's the small things that show why this episode, and the series as a whole is so great. It paces back and forth between the comical and serious, and that's thanks to Spike.
He is the reason you watch the show. Between his good guy vibe and his lethal skills, you couldn't ask for a more bad ass hero. Spike is a bounty hunter, a good one, but he doesn't come off cocky or sadistic. He just seems like a nice guy who understands his role in life and goes with it. Charming is the best word to describe him, and it comes across when he interacts with his target, Katrina. And as the chase continues, and the situation looks more bleak, Spike continues to realize that this is his job. No matter how ugly it gets, he has to do what he has to do.
I won't go into great detail about what happens, I'll let you decide that for yourself. But all I can say is that when Katrina utters her last words and the battle is over, you can't help but feel the way Spike does. He's ambivalent about his lot in life, but like a spoke on a wheel, he continues because he knows that's his role in life. Few shows, let alone animes, can pull you in so closely and leave such an impression with it's opening episode.