Good animes..... its really up to taste.
Its difficult to say what I like is something you are going to like. I've got weeeird tastes. Then again I see anime like any other television programs, so I can have eclectic tastes. For some reason I like BlackJack alot... which is a really good series of movies. Its premise was good enough to draw me in (its about a doctor). It was also based on a manga by Tezuka Osamu, who we should all know is the father of japanese animation and manga. Somebody on another thread talked about slam dunk... I think thats a decent series from what I have seen.
One of my favorites is Umi Ga Kikoeru... or I can hear the sea. Its a ghibli affiliate. no action or big events, its about a love triangle between two friends and how they deal with it when they return for a reunion. Its exceptionally hard to find and only an hour long.
To be honest ANY Studio ghibili film is a must see. If you haven't seen Nausicaa and/or Laputa, you are missing out.
The Wings of Honneamise... gainiax's first film... its a virtual standalone great anime. Probably some of the best animation not just technically but also conceptually. The world itself is an alternate universe very similar to our own 1940s/50s. But things are just designed that slight bit differently that you notice the change. Like coins that are metal rods, rather than discs or language or signs or trains. But Gainiax did this because it says, no matter how things look differently, humans are all the same underneath. Its about this world's first attempt to launch a human into space. It kinda gets preachy through the middle, but some of the scenes are spectacular, as well as the story.
Wings can only be eclipsed by one movie... so utterly shattering that I couldn't watch or read anything for the next few days... Hotaru No Haka (grave of fireflies). Heartbreaking and poignant... it may be one of the deepest films you will ever watch. actually Its so good that I would warn people from watching it. Its not the violence or anything likethat, it will just ruin your day.
On the lighter side, Ah megamisama (ah my goddess) is really the precursor series to a lot of the stuff like Chobits and Onegai Sensei. Yes there are other simpler series out there before, but AM was the first to make the animation suit it, keep the episode count low and semi serious comedy. And no matter what you say Bellandy is the original super goddess of anime.
An even better series is Lupin III. Its about the grandson of the famous theif Arsean Lupin, and his capers to steal stuff. There are five elements in every episode and movie. Lupin, the comedic main character, jigen his mafioso american cigarette smoking sidekick, Goemon greatest japanese swordsman ever, Hot Girl theif/antagonist (usually Mine Fujiko) that lupin drools over that will proabably get his loot half the time when he is drooling over him that Jigen always dislikes, Evil Manichal individual like goldfinger, and Inspector Zenigata Keibu, who has devoted his life to capturing lupin, always Lupin outsmarts him.
Now I'm going to state that a lot of animes I have seen and have NO CLUE WHAT THE NAME IS. There is one I have seen in Japan, its kinda like aliens, but in the end only two people survive when they take a shuttle down, and the station explodes when one of the main characters lights his lighter in the H2 room. Somebody tell me what the name of this movie is...
On similar lines, I think I have seen gasakari... I was reading anime lists when writing this thread, and I read its synopsis... I remember watching a series exactly like it.
There are so many other animes that I could say, but I'll leave it at that. I hope this helps