The series I would recommend as being classics (in both the 'established classic' and the 'instant classic' meaning of the word) would be:
Oh My Goddess! (OVA)
The movie, TV series, and animated shorts go under the title Ah! My Goddess; highly recommended.
Robotech
This is actually one of the earliest hugely popular anime in America. It's actually three totally unrelated series - Macross, Southern Cross, and Genesis Climber Mospeada - that were combined and the plot changed for continuity. Unlike the hostility that Cardcaptors or Sailor Moon met when the English versions of those series came out in America, Robotech isn't considered to be a 'butchering', unless you talk to rabid Macross fans. It did run on Toonami seven years ago, and only reappeared on Giant Robot Week which was still about two years ago. Macross itself has several spin-offs, such as Macross Plus, Macross 7, and Macross Zero (which is the newest, and finished getting released sometime last year, I think)
Ranma ½
This is the Rumiko Takahashi series that immediately preceded Inuyasha (actually, I think the release of beginning of the Inuyasha manga and the end of Ranma ½ series franchise overlapped; this of course, is referring to the Japanese release)
Magic Knight Rayearth
This was produced by CLAMP, whose other works include Cardcaptor Sakura, Chobits (also recommended), X (recommended with discretion), Angelic Layer, and RG Veda, among others.
Neon Genesis Evangelion
This is often referred to as 'the greatest anime series of all time'. That's really up to personal opinion, but it did influence a massive amount of anime fans and the industry itself during the mid-90s. Interest in the series was revitalized by the 2003 release of Evangelion 'Renewal' (known as 'Platinum' in the States, due to ADV Films not having the licenses to the movies; Platinum is in the process of being fully released here), a total remastering of the series. The overall quality change is just unbelievable. Look at this comparison (the original is on the left, Renewal/Platinum is on the right):
Another thing to note, though: the Evangelion manga is not exactly the same as the series. The series is Hideaki Anno's vision (Anno was the director of the series), and tends to be more psychotic, while the manga is Yoshiyuki Sadamoto's vision (Sadamoto was the Character Designer for the series; this gives the manga the added bonus of it's animation being uniform, unlike the series), and tends to be more emotionally involved, not to mention that certain events don't happen quite the way they did in the TV show. I prefer the manga, but the series is still one of my absolute favorites.
Love Hina
Another series that I prefer the manga over the TV show (due to the TV show seeming to be a bit rushed), but highly recommended. A third-year college hopeful is placed in charge of his grandmother's all-female dorm. Wackiness ensues. It's one of your typical romance series (sometimes described as a 'harem' series, meaning 'one guy, bunch of girls'). The anime is recommended for 13+, the manga for 15+, I believe.
Another series from Ken Akamatsu that is somewhat similar in theme is Mahou Sensei Negima, which is sometimes described as 'Harem Potter', due to it's lead character - a 10-year-old boy magician placed in charge of an all-female 8th grade English class. This series is currently being fansubbed by AnimeForever and AnimeCouncil, along with about four or five other fansub groups. The manga is already being released in English.
Please, Teacher!
This is a series that I believe would fall under the 'instant classic' category. An 18-year-old high school student (who looks 15 and is in the 10th grade because of a debilitating disease) is forced to marry his extraterrestrial teacher due to a misunderstanding by the school principal. The two have to hide their relationship from his friends and classmates. Despite the odd circumstances (which many are uncomfortable with due to the creepiness factor of teacher-student relationships), this series actually addresses it's main issues (growing up, first and foremost) in a very mature and respectful manner. The AMV I'm advertising in my signature is a Please, Teacher! video.
The 13th episode is the lone exception; it's just pure fanservice. There is a sequel series, Please Twins!, but like the 13th episode, also has a large amount of fanservice. The difference, though, is that Please Twins! actually has some maturity it does deal with aside from the fanservice, so that only barely makes up for it. The original series' main 12 episodes are highly recommended, if you can get past the idea of a high school student marrying his teacher.
Jinki: Extend
Another one of your seemingly typical mech series, this is actually a sequel to another manga, entitled Jinki, that for one reason or another hasn't been licensed for release in the States (even though it's sequel manga is *rollseyes*). The anime version of the series is currently running in Japan and being fansubbed by ANBU.
. But I don't know much about it. As much as I want to buy a lot of books to figure this out on my own, and watch the cartoons...Well the books are kinda pricey, (Don't want to be into too many stories...At most 10 haha) and as for the TV goes all I've got for that is Cartoon Network so...that's a bust. ( I've got dish on demand ) So here is where you lovely people come in. I would like you great people to show me the road to being a true Anime person...Not a lame 'poser' 
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