I have assumed that the TV show was rather silly, (kinda liked the movie), but mainly because I assume most TV is rather silly. However, I have just finished watching Firefly and Serenity and find them, frankly, to be totally awesome. This Wheedon guy is good. So, sell me on Buffy…
If you like Firefly and Angel, odds are you’ll like Buffy.
Only after posting do I realize you said Firefly and Serenity, not Angel. Man, my reading comprehension skills are awesome today.
Sigh…
Really, the best selling point is in watching the series itself - see if you can borrow a season 1 DVD from someone, or catch it on TV.
I can Netflix it. I am wondering if it has as much…weight as Firefly.
Hmm. I would say yes, but it’s a different kind of weight. In Firefly, it’s usually moral and personal issues that are explored. In Buffy, there’s generally less of the moral (after all, it is very much more good guys vs. evil demons, at least in the first few series) but the personal issues remain. Also, since the main characters are teenagers, there’s distinct “coming of age” storylines and emphasis, which has a very strong effect on character development; the Buffy in the first episode, for example, is very, very different from the Buffy in the last - and even from the Buffy in the last episode of the first season.
The “reality” also remains, if that makes sense for a fictional show. In Firefly you get the sense that yes, if reality really was like that, people really would act like that - you get the same sense from Buffy. They may be killing demons and vampires, but the reactions and personality of the main characters do seem understandable and true to life (mostly).
I’m sure a more well-versed Buffy fan will be along in a minute (there seems to be rather a lot of us on here 
) who’ll be able to speak about this.
Sometimes it certainly does, sometimes more, even. Of course the setting (young people in high school and later college) has an effect on the tone–which is often kind of deliberately campy. The emotional resonance the show creates, however, is absoluting as affecting and sophisticated as Firefly; at least it is to me. Joss Whedon uses the campy, teen-horror setting as a point of departure in metaphor, instilling the show with a compelling mix of humor, drama, and horror. Fundamentally it’s all about the characters: how they live and grow as human beings (in this fantasy setting) where moral choices become increasingly difficult. The show, again similarly to Firefly, rewards the viewer who sticks with the show for multiple episodes. The larger plot arcs are always driven by character development.
Yeah. Sell me on it, too. I love Firefly, but I’ve never seen Buffy.
Like you, I’ve seen (and loved) Firefly but hadn’t seen Buffy.
A friend of ours lent us the first series, which we have now watched. I don’t love Buffy. I’ll be happy to have watched each episode once, whereas I’m still replaying Firefly.
I find the high school setting of Buffy, with it’s ‘cool kids’ and ‘uncool kids’, kinda cliche and the teen angst just doesn’t appeal to me. YMMV.
As one of the biggest Buffy and Angel fans on the SDMB, I have to warn you: BE PATIENT. When a friend first thrust the Buffy season 1 DVDs into my hands about a year ago at this time, I was extremely skeptical. I didn’t think I would like it, and it took me a while to get into it. I decided I loved it by the season 1 finale (and season 1 is a short season of only 12 episodes), but please don’t give up too soon. The show is very low-budget, especially compared to Firefly and Serenity, but the monsters and demons and special effects always take second stage to the witty dialogue, character development, and wonderful acting. The show gets better, and most fans agree seasons 2 and 3 are the best of all.
If you stay loyal and watch the DVDs in order, the Angel series spun off when Buffy entered its fourth season. I think Angel season 1 was hit-or-miss (with some very excellent hits), but that show only got better and better, culminating with the excellent fifth and final season. Most fans agree Buffy went on too long and wasn’t nearly as good by its final (seventh) season, but I enjoy the later seasons more than most. I implore you to give it a chance, though! If you have the slightest inkling you’ll like it, you probably will. And unlike you guys who already know Whedon’s work from Firefly, watching Buffy on DVD last year was my first exposure to the man’s style. You’re in for a real treat, all of you!
The general consensus–which I agree with–is that Buffy improved from season to season; it really did just keep getting better. The corollary of that of course is that the first season is the weakest.
Keep going.
Then there is the sheer rapture of watching Buffy Season 4 alternately with *Angel * Season 1, and seeing the episodes they linked (they were on the air back-to-back that season). Nothing like thinking you know something because of the way it way presented on Buffy, and then seeing another perspective on Angel.
Buffy was the first, and it shows it, for both good and bad. The things that were good were so very good. Fresh and new. The things that were bad were…still pretty good. 
Seeing it for the first time at this remove means you won’t have to take sides in the Shipper Wars. They made the Thirty Years War seem like a child’s picnic. Oh, the Humanity!
Besides, where else will you find dialogue like:
Willow: “Buffy, earlier you agreed with me about Thanksgiving. It’s a sham. It’s all about death.”
Buffy: “It is a sham. But it’s a sham with yams. It’s a yam sham.”
(In the interest of full disclosure, I sided with the Spike camp.)
The wit in the dialogue alone sets Buffy apart from most of the crap being shown at its time, and makes the show worth watching. But especially early in the show, the extent to which they manage to make a theme play out on multiple levels of a story and reify the internal demons we all struggled with mentally as adolescents into actual demons that Buffy and her friends have to struggle against physically creates what for a lot of people is a powerful resonance with the series.
Not to mention:
*Anya: I love a ritual sacrifice.
Buffy: It’s not really a one of those.
Anya: To commemorate a past event, you kill and eat an animal. It’s a ritual sacrifice, with pie. *
Watch it through mid-season 2, specifically the two parter Surprise/Innocence. If you’re not convinced by then, you won’t be.*
*And you’ll know there’s something wrong with you.
Definitely give Angel a try as well. The characters and storylines on Angel are a bit more adult, and I find that as someone who came to the Buffyverse already well beyond high school, it sometimes resonates for me in a way Buffy simply doesn’t, no matter how I love it.
“Just a doodle. I do doodle. You too, you do doodle too.” 
What? S6 is one of my favorites and I like S7 well enough but you and I seem to be in the distinct minority on those points.
No kidding. Not Fade Away hit me much harder than The Gift or The Chosen did.
“Let’s get to work.”
Brrrrr.
“Well personally, I kind of want to slay the dragon.”
*Spike: What do you think all this means for that Shanshu bugaboo? If we make it through this, does one of us get to be a real boy?
Angel: Who you kidding? We’re not gonna make it through.
Spike: Well, long as it’s not you. *
See what you started, Contra? If you aren’t careful, we’ll do this all night.
“How come the sudden calisthenics? Aren’t you sort of naturally buff, Buff? [laughs] Buff Buff!”
I less-than-three Willow.