No, not all of them are sci-fi. There's plenty of other genres - fantasy, romance, action, sit-com, so-called 'slice of life' series (which tend to be either school or harem comedies with romance elements thrown in), drama, cop show, etc. There are even shows which specifically set out to parody all of the aforementioned genres.
Seeing as how I've never seen Spirited Away or Princess Mononoke, I don't really know what to recommend that's similar to those.
It depends on what sorts of things you're comfortable with. My suggestions run from light fare to mature series:
Light: nothing that would really be considered objectionable
Semi-light: very little that would be considered objectionable; may contain some stronger language or very occasional nudity/semi-nudity in a comedic context (transformation sequences that involve light nudity would also fall into this category)
Medium: a more moderate amount of things that could be objectionable; may contain more frequent stronger language or nudity/semi-nudity used in both comedic and provocative contexts, some light sexual situations; violence is mainly slapstick in nature, and not very graphic
Semi-mature: probably contain things which are questionable, presented in a comedic or darkly serious context; violence is more gratuitous, in both slapstick and serious contexts; sexual situations more pronounced, possibly on par with mid-day soap operas, nudity probably used more for provocativeness than comedy.
Mature: contains things not for younger audiences, but not inappropriate contentographic - about on par with R-rated films; violence can be very gratuitous, strong sexual situations or strong innuendo, high likelihood for very serious plotlines, dark psychological plots, and other various things.
Ah! My Goddess (semi-light; romance/comedy, fantasy) - I personally recommend the
miniseries (OAV or OVA=miniseries), but there's also a
movie, and most recently two television seasons (
first season, and
second season - the second season's American release date I'm currently unaware of, although it has been licensed for U.S. distribution; the first season has already been released)
Elfen Lied (mature; action, psychological, horror)
Robotech (semi-light; action, space drama, sci-fi - the first season of the show is actually the Americanized version of the
original Macross series)
Neon Genesis Evangelion (semi-mature; extremely psychological in later episodes, some sit-com elements early on, giant robot sci-fi, very liberal use of religious imagery for flavoring)
Azumanga Daioh (light; school sit-com)
School Rumble (medium; school sit-com, romance/comedy) - currently only available in English in fansub form, currently in its
second season, there's also a
2-part miniseries that fits between the two seasons
RahXephon (semi-mature; action, giant robot sci-fi, psychological, romance - often considered as a less intense version of Evangelion, Mayan imagery used in place of the Judeo-Christian imagery in Eva)
Please Teacher! (semi-mature to mature; romance, comedy, drama, some light sci-fi elements)
The Melancholy of Haruhi Suzumiya (semi-light; psychological, sci-fi/fantasy, light comedy/drama) - currently only available in English in fansub form
Bleach (semi-mature; action, fantasy, comedy; centrally involves the concept of Shinigami - literally 'death god', in actuality more of a Grim Reaper/Angel of Death sort of figure) - release date in America unknown; will be premiering on Cartoon Network's Adult Swim block in September
Kimi Ga Nozomu Eien (mature; romance, drama) - release date in America slated for Winter 2006
Mahoromatic (mature; sci-fi, somewhat-harem comedy, romance) - there's also a
second season and a
Summer Special (which chronologically occurs at some point during the second season, although I would recommend watching it last to soften the blow from the final episodes)
Galaxy Angel (light; sci-fi, comedy, action; Looney Tunes-style slapstick sit-com) - has four seasons, and a to be announced sequel television series.
In general, the
Anime News Network Encyclopedia is a good place to look up information on a series.