@RekkaAtsuma:
Silver Spoon (Gin no Saji). It's about a city kid sent to an agricultural boarding school in the country, and having to learn all the shit about animal husbandry. A real eye opener about where meat comes from
Beck (mostly drama oriented, a little comedy, it's about a kid who learns guitar and forms a band with some friends, he's 14 at the start of the series)
Bakuman: written by the death note authors, it's about 2 guys who create manga. don't want to spoil too much, but it really goes int detail about how the manga business works at every level, and has some great characters. It starts when the main characters are 14 y/o, and ends when they're ~22, so the school stuff is sort of a side story here.
Ensemble comedies (mostly they're about a group of same-sex friends) include:
Azumanga Daioh (probably the template for most "cute girls in a classroom" series, but a lot better than most imitators.
Hidamari Sketch (trippy SHAFT artwork version of the cute girls concept, really endearing characters)
Acchi Kocchi (kind of a cute overload)
K-On (kind of infamous as the show about a cute schoolgirl rock band who do almost nothing but eat sweets and drink tea and chat about nothing.
occasional song performances, once in a blue moon)
Kimi to Boku (a rare one about a group of male highschool friends that isn't smutty)
School comedy and/or drama with romance include:
Kimi ni Todoke Toradora Sakimachi no Appollon (1960's setting, about jazz muisicians, great soundtrack, drama and romance subplots)
Shigatsu wa Kimi no Uso (Currently airing, it's similar to the above but about classical music + school & romance)
Gekkan Shoujo Nozaki-kun (just finished airing, it's a comedy about a group of students who write shoujo manga (manga for girls, usually with cheesy cliched romance plots) and many of the characters and incidents are parodies of shoujo manga conventions.
There are a fair bunch of college-age slice of life drama/comedy/romance series as well which don't have any supernatural element, e.g.
Nana
Nodame Cantabile
Honey and Clover
FAKEEDIT: aenri beat me to the post button, consider anything on both lists to be highly recced. I should watch a couple of aenri's suggestions too that I haven't seen.
BTW I've seen Genshiken but forgot about it. It starts in college though, the entire concept / focus is the club called Genshiken in the college. In the first episode the MC joins the club. Maybe highschool is a flashback or something, but it's definitely not the start of the anime.