I got into anime around the same time everyone else in my age group did, when Dragon Ball Z started airing on Cartoon Network. I don't know if I could honestly call myself an anime fan, since it is a medium, and not a genre, but I have watched quite a lot that I have enjoyed.
I've also watched more than a few that have ranged from insipid to insulting; I get most of my recommendations from friends, and their tastes typically include multiple moes molesting main men. :P
Here's a little list I've compiled of all the anime that I have seen from start to finish:
Top 3
Trigun
Welcome to the N.H.K.
Cowboy Bebop
The Rest
Attack on Titan
Kill la Kill
Tenchi Muyo OVA (All three seasons)
Tenchi Universe / TV Tenchi!
Gundam Wing
Death Note
WataMote / “It's Not My Fault I'm Not Popular”
Haganai / Boku Wa Tomodachi ga Sukunai
Puella Magi Modoka Magicka
Shuffle!
Iron Man
Air Gear
Now and Then, Here and There
Persona 4: The Animation
Shit Tier
School Days (Guilty pleasure; nice boat!)
Clannad (god fucking awful)
Hoshizora e Kakuru Hashi (barely even remember it)
Movies
Tokyo Godfathers (Simply the best)
Howl's Moving Castle
Summer Wars
The Girl Who Leapt Through Time
Princess Mononoke
Spirited Away
The Castle of Cagliostro
Akira
Trigun: Badlands Rumble
Cowboy Bebop: The Movie
Persona 3 The Movie: #1 Spring of Birth
Metropolis
Tenchi the Movie: Tenchi in Love
Tenchi the Movie 2: The Daughter of Darkness
Tenchi Forever! The Movie
Currently Watching
Excel Saga
JoJo's Bizarre Adventure (2012)
... and that about sums it up, besides the animes I started watching but gave up partway through on.
I really don't watch that much anime. HOWEVER, I did love Tokyo Godfathers and Redline. The only anime show I've watched in the last couple of months is Watamote. It's painfully funny to watch.
Welcome to The N.H.K. was a great show.
You two have good taste, and I commend you.
Ahh, I need to finish watching season 2 of that some day. I had a lot of fun with the first season.