I recommend "Kimi ni Todoke" over "School Days", any day. I refuse to mistake angst for romance.
"Kare Kano" is licensed as "His and Her Circumstances" in the US.
Vandread has a kind of romance to it. Mostly it's sci-fi shounen, but...well people fall in love in it.
Urusei Yatsura Those Crazy Aliens, was a mainstream comedy, but was based on a "marriage" between an earth male high school student and an Oni princess from another planet.
Kanon, it's very like Clannad, but up north. Both are wonderful anime.
Karin, known as Chibi Vampire in the manga. Comedy but a romantic comedy. Ask Superjaz about it, and why she is in the manga.
You could call Chobits a romance. Love it.
Midori Days, another romantic comedy.
Then you have Voices From a Distant Star, and The Place Promised in Our Early Days. Beautiful excellently done anime. Yeah, they're masterpieces.
One of my favorite anime is 'The Story of Saiunkoku'. It has romance, comedy, political intrigue, great art, nice music, and a good plot. At least I think so. It takes place in a different world that looks a lot like ancient China with an Emperor, the royal court, and noblemen. It can be a little slow in places, but there is action, and while romance isn't the main point of the series it is one of the driving point between the main characters.
Anther one I haven't seen mentioned is Ai Yori Aoshi. It's a romantic comedy, and while it has a kind of a harem thing going, along with a fair arount of fan service, the main characters really are cute together.
I agree with both of you.
I will recommend
Skip Beat. This is mainly about a girl who gets dumped by her boyfriend and decides to get revenge by becoming a more popular pop idol than he is. She also ends up falling in love with a better person as the show progresses. A great comedy, romance. I highly recommend it.
I also liked
Ginban Kaleidoscope. An anime about a female figure skater who gets possessed by the ghost of a male stunt pilot. Sort of like the movie
Ghost. A tragic romance comedy since the guy she falls in love with is, of course, the ghost who is possessing her.
No one mentioned
Ah My Goddess yet. The romance of a college boy and a goddess. It is mostly a comedy, a bit of harem, although Belldandy is the only girl he loves.
Millennium Actress - This movie is all about unrequited love. By the late Satoshi Kon, we will miss you.
Amagami SS - This anime is based off of a dating sim and plays out like one. Each of the possible girls Tachibana, the main protagonist, can be with has a four episode story arc. This anime is all about romance, The best part is, that no matter which girl the viewer likes, there is a story arc that has him with her.
Lovely Complex - This one is just plain fun. A romance between two students who are so alike in attitude, but different in everything else. She is tall, very tall, and very self conscious of it. She wishes she was short and cute like most of her classmates and friends. He is short, very short, and wishes he was tall like most of the other boys. The two hate each other, because each has what the other desires. Of course they are always arguing and fighting, and of course they find themselves liking one another.
As
Jaybug stated
His and Her Circumstances aka Kare Kano - This story is fun to watch, but I often think the VA musty have gotten hoarse. She narrates constantly. Similar to
Lovely Complex in that the main characters start out not liking one another and then slowly begin to fall in love.
Special A - I think of
Host Club when I watch this one. The girl has one desire, to beat the boy in anything. Problem is, she never does. The boy has one desire, to have the girl realize that he likes her. They are totally hopeless.
Cross Game - I put this here as an overall example of a great anime story that happens to be about baseball. The real story has nothing to do with baseball though. If you like romances, you may enjoy this show. It is definitely a soap opera. One of my all time favorite anime.
That should get you started.