Wow, I really admire your ambitious goals. I wish you the best of luck in creating something like that. It sounds like you're trying to appeal to a wide range of people without compromising your story's original feel, and that sounds like the kind of thing I'd want to see done.

I don't think you need to be an over-the-top funny person to produce an entertaining show. Of course, easy ways to get laughs are slapstick (way overdone), non-sequitir (which can kill a serious mood and throw off the progression of the story), and maybe wordplay (actually, I happen to be a sucker for puns), but they're not the only ways. Situations can be amusing without explicitly using "jokes."

In my opinion, one of the most effective types of humor is the kind that's "funny 'cause it's true." Things that are relatable to audiences are likely to interest them. It's a kind of humor that would be easy to write and still flow nicely with the story.

The only thing I can think of to explain that appeal is the anime Lucky Star, which I realize is not the kind of model you're following for your own show, but is really a good example of simple but effective humor. Setting aside the jokes that derive from fanboy-recognizable shout-outs and references, most of the comedy in the show is a result of the honest characterization and lighthearted portrayal of everyday situations. Whether it's an innocent friend posing an uncomfortable question ("Tamura-san! What's yaoi?"), an internet connection being lost after an hour of typing ("No! Back, back!" *page has expired*), or losing track of time while exerting an absurd amount of brainpower on a trivial subject ("So, which end of a chocolate cornet do you start with...?"), there's something funny about seeing lively, animated faces experiencing the same kinds of things as you have. It also accomplishes the realistic-yet-cartoonish vibe that you want to get across. Just being honest and straightforward works wonders. Even if everything in the story turns out to be "boring," poking fun at that in and of itself might make it humorous.

Another idea is to create some sort of overhanging irony with one of the characters, and have them be constantly reminded of it. I can't think of an example, but... yeah... Something where the situation isn't exactly funny, but strange in some sort of way that grabs the audience's interest... I don't know. But along the same tangent, maybe there could be some sort of quirk that the protagonist has that is constantly yet unintentionally comedic?

If nothing else, you could introduce some sort of comic relief character. An animal companion, an overly energetic tagalong, or whatever - while the main protagonists don't get their serious disposition corrupted, there can be a minor character that helps keep the attention.

In any case, you should by no means try to force humor in the story. I think, if you let the story take its course, you might be able to find the subtle humor in something and discover a way to amplify it.

I'm an avid observer of animation, but I'm not very good at either drawing or creating original ideas myself, so I'm sorry if I'm not much help. But I do hope you find a way to make it all work out. ^^

EDIT: Ah. I totally missed the part about the actual animation. I'm currently on a computer that doesn't allow me access to a lot of good image sites, but when I get back on my normal computer I'll try to find examples of styles that I think look believable without being too dull or realistic. I think of it like... animation should take you into another world that mimics life. So it's realistic, yet other-wordly. It's an escape from real life, yet somehow familiar. The more mature sorts of anime that don't use large eyes or facefaults are good models.

EDIT AGAIN:
http://upload.wikimedia.org/wikipedia/en/thumb/1/11/Haruhi_book_01_s.jpg/402px-Haruhi_book_01_s.jpg
The Melancholy of Haruhi Suzumiya had no facefaults throughout the whole series. Even while tackling humorous material, it kept its art looking believable and respectable.
http://media.photobucket.com/image/shakugan/dougportillo/Shakugan%20no%20Shana/Shakugan-No-Shana_.jpg
And Shakugan no Shana was from the same character designer.
http://www.sundevildvd.com/store/files/t_17463.jpg
Kino's Journey effectively maintained a very serious mellow feel in its graphics to reflect its often solemn tone.
And... I just realized that I mostly watch things with brighter and crazier animation, so that's all I've got.... But yeah, there are a lot of ways to make a style appealing in a more mature and realistic way - not too boring but not too cartoonish, and, on the flip side, cartoonish but not too wild. I'm sure your character designs will reflect however you plan for them to look and act throughout the story; animation technique usually seems to adapt to the mood of a series.