General Discussion

@ChristinaAV: 

I've seen a bit of the Slice-of-Life genre although my focus leans toward romantic comedies.  To begin with, you should seek out Aishiteruze Baby.  It's about a teenage boy who takes care of his 5 year-old cousin when she's abandoned by her mother.  I found it very heartwarming and also a bit melancholic.  A few others you will want to look into are:
  
Soredemo Machi wa Mawatteiru
Minami-Ke (3 seasons + OVA) 
Hidamari Sketch (4 seasons + OVAs)
GA Geijutsuka Art Design Class
Sora no Manimani
Ichigo Mashimaro (aka Strawberry Marshmellow)
Kyou no Go no Ni
 
A-Channel was the last one I finished and if you liked Lucky Star it's a lot like that although without the excessive but witty dialogue.  Lucky Star's trick was that they'd spend half an episode talking about the proper way to eat a cream-filled pastry and A-Channel has some of that celebration of the mundane, but it lacks the commitment that LS took.  Still, I enjoyed it since I liked LS. 
 
Currently I am watching one called Binbou Shimai Monogatari which is about two orphaned sisters who are very poor and do odd jobs to survive.

For something that's not exactly Slice-of-Life but still "anime that's not too anime", I think you might like Arakawa Under the Bridge (2 seasons), Ghost Hunt, So Ra no Wo To and my personal all-time favorite show, School Rumble (2 seasons + OVAs).

@ChristinaAV:  Well, School Rumble is a romantic comedy so that would be my first suggestion.  It's about a school delinquent who falls for a perky girl who is in love with a bland quite guy who barely knows she exists.   The fun doesn't stop there as the series weaves a web of inter-romantic relationships between the large cast.  The first two seasons are 26 episodes each and the final is sadly just two OVAs that pretend to be a full season.  It's a bit of a let down, but it provides some needed closure just the same. 
 
After that, I'd second @sotyfan16's recommendations of Toradora and Kaichou no Maid-sama.  The former is about a very fiery but petite girl who learns to love a guy who looks mean but isn't.  It's one of the best romantic comedies ever.  The second half of the series is more serious though.  Maid-sama is one of the more recent one's I've seen.  It tells the story of a poor student body president who secretly works at a maid cafe to help out with her family's living expenses.  He father ran off leaving herself, her mother and younger sister to survive on their own.  A mysterious guy with a wry sense of humor is the male love interest. 
 
If you want something that's wacky and much more comedic, I'd suggest Seto no Hanayome (aka My Bride is a Mermaid).  In this tale, a normal high school boy accidentally gets engaged to a mermaid.  If that wasn't problematic enough, her father is head of the mermaid yakuza/mafia!  Another funny one is Baka to Test to Shokanjou (aka Idiots and Tests: Summon the Beast).  It's about a guy in the lowest level of education at his magic school and a girl what happens when a girl who should be in the super-advanced winds up in his class.  The second season is currently airing on Japanese TV.   If you don't mind something that's a little more perverse, you might take a look at B Gata H Kei.  The premise is a girl wants to have 100 sex partners but winds up having difficultly relating to just one potential interest.  The humor can be a bit lewd, but the story is really about the confusion of first love.   There's nothing obscene in it, too. 
 
A more traditional one I saw recently is Lovely Complex.  It's about a very tall girl who falls for a short boy.  They both have height issues and so their love is difficult.  I liked it a lot although I think it spent a tad too much time in indecision.  Still, a heartfelt show worth seeking out.  Another is Amagami SS, which is a series of short love stories in which one male character takes several different paths to love several different girls.  It's kind of like each girl's story is a separate timeline.  I'm half way through the show and I like it but shorter tales mean less time developing a connection to the characters in my opinion.   
 
Finally, two shows that aren't primarily romantic comedies but have a substantial amount of romance in them are Eureka Seven and Nadia: Secret of the Blue Water.  They're similar to each other in that they are about two young kids falling in love while on a crazy adventure aboard a large ship.  I really enjoyed them and E7 made me a bit misty at the end.  Be warned though, they might be more anime-ish than you're looking for.  Eureka has giant robots and Nadia is a steampunk story based off 20,000 Leagues Under the Sea. 
 
I keep a constantly updated list of harem and romantic comedies I've seen here on the site.  I don't go into too much detail there, but I'm thinking of fleshing each out a bit more.  I didn't bother to suggest any harem-comedies here, but the ones I've enjoyed are listed there, too.  Harem and Romantic Comedies I've Enjoyed a Lot, Somewhat, and Just a Small Bit
 
EDIT:  Urusei Yatsura and Tenshi na Konamaiki (aka A Cheeky Angel) are two other favorites of mine, but I wasn't sure if I should mention them.  The former is a classic screwball rom-com from the early 1980's that might not appeal to modern tastes.  I really love it but I love old anime, so I might be unlike the average viewer.  The other one is a reverse harem, meaning it's about one girl pursued by several guys.  Even with that format, it's really about one guy and one girl...well, the girl might have been magically transformed from being a boy originally.  She's trying to find the genie that made her a girl so she can switch back, but love could find her before that.  The animation isn't too great in this one since it reflects the design from the manga, but I still really enjoyed the story.

@ChristinaAV: I second the choices for Lovely Complex, Lucky Star and Ouran High School Host Club. 
 
If you want something more on the dark and serious side, then I recommend Phantom: Requiem for the Phantom and one of the newer series, Madoka Magica, but I suggest you go see the first two Pretty Cure series (the original and Max Heart) before Madoka Magica for familiarity within the Magical Girl genre even though Pretty Cure's not as dark as the former.