Good Slice of Life or fantasy anime?
 

I already know about K-On! Lucky Star, etc.. any more recommendations?

For fantasy, I'm looking for something like Angel Beats and The Melancholy of Haruhi Suzumiya...you know the alternate world kind of thing.

Kamichu! - slice of life and fantasy, a very good anime

For First I'd say Seitokai no Ichizon for sure. And maybe some of these series Working!!, Sora no Woto, and Sora no Manimani for Slice of Life comedy like those.

And for the second maybe Bakemonogatari, it's only series I can think of like them in the sense of comedy and being bizarre, yet it's still very different from them :p. Though technically it would be more supernatural than fantasy, but I'd also say supernatural fits both series you named batter than 'fantasy'. Most other supernatural or fantasy series I would consider recommending general lack the same kind of feeling, and use of comedy that's present in Angel Beats, and Haruhi Suzumiya. And tend to be more action focused.

Aria is pretty much the definitive slice of life series. While it isn't as comedic as Lucky Star or K-ON!, it's just as feel good and heartwarming (if not more so) as those series are.

Minami-ke- A comedic slice of life series. The first season is the best and the funniest. Different from many others in the genre in that it has more prominent and memorable male characters, though the leads are female.

Slice of Life

• Azumanga Daioh - High school girls day-to-day, ridiculous humor, pure awesomeness
• I second Aria - mainly girls again who are gondoliers ("Undine") in a city titled Neo-Venezia on the planet Aqua. There are three wonderful seasons: Aria the Animation, Aria the Natural, and Aria the Origination.
• I also second Minami-ke--About a family of 3 sisters and their home/school lives. There are also 3 seasons, with slightly different graphics for each one. Most people agree on the 1st and 3rd seasons being the better, with the 1st as the definite best.
• Ichigo Mashimaro--The home/school life of 4 elementary girls
• Hidamari Sketch--About girls who go to high school Art School. 3 wonderful seasons ^_^ Great things come in 3s!
• Another one previously mentioned I agree with is Sora no Woto--Art style like K-On!, military girls, a somewhat beautiful/Europe setting post-apolyptic world. Also focuses somewhat on music, though the connection is fairly loose.

Fantasy

• Mushishi--Episodic series following Ginko, a Mushishi. He studies the phenomenon of Mushi, a lifeform beneath insects, even more difficult to see than microscopic organisms. They seem to balance between the real world and the spiritual. If you like Aria, you'll most definitely like mushishi, which has a real dreamy/myterious sort-of feel to it.
• Kamichu!--blend of Fantasy and Slice of Life as previously stated. Yurie, the teenage goddess (really middle-school) goes through this episodic series trying to figure out exactly what kind of god/middle schooler she's supposed to be. Very cute show.
• Natsume Yuujinchou--2 seasons of this greatness. Also a bit of mix w/ slice of life. Natsume can see the spirit world. The episodic series traces his different encounters with spirits based largely on a book containing the names and powers of many spirits defeated by his grandmother.

Honestly...there are too many to name!

You could try fruits basket( I dont know how many times i mention this anime XD) Its in between both in a way

All the shows you mentioned were comedies. I'm going to toss out a couple of other series about "the alternate world kind of thing." These are more dramatic though they all have their share of comedy.

First, there's Noein, where three alternate worlds are in conflict. It's an exciting science-fiction story, though one built on rather shaky interpretations of quantum physics. You won't care about the accuracy of the science because you'll be wrapped up in the remarkable animation and warm-hearted story.

Next you might try Dennou Coil, another sci-fi story set in the near future. In the "dennou" ("computer") world, kids wear special glasses that let them participate in a virtual world that's overlaid on the physical one. It's one of the best series of the past decade and has a wonderful take on the "alternate universe" type of story. Again there are hilarious segments and some very sad ones.

There are also more traditional fantasy stories like The Twelve Kingdoms where a wimpy Japanese school girl is magically whisked into another world and discovers her true calling. You need to watch the first half-dozen episodes before deciding whether to continue, though. Wait until a rather special character first appears.

If you like Twelve Kingdoms, or even if you don't, you might want to give Moribito or Kemono no Sou-ja Erin a try. Like Twelve Kingdoms, these two series are adapted from "light" novels.

As always, Marina's list shows her excellent taste!

I agree with, Kamichu!, Working!!, Bakemonogatari and highly recommend Aria as it is an extremely well done SoL.

For a parallel universe/world anime i'd recommend the recent OVA Black Rock Shooter, the first Fullmetal Alchemist, and a bit of Inuyasha