I've read and watched TOO many to count @_@ since I've been obsessed with manga since I was in second grade. But here are some that I remember reading/watching:
Shounen manga:
Dragonball
Rurouni Kenshin
Naruto
Bleach
Yakitate! Japan
Shaman King
Detective Conan
Busou Renkin
One Piece
Inuyasha
Full Metal Alchemist
Full Metal Panic! series
Samurai Deeper Kyo
Prince of Tennis
Digimon
Pokémon Special (Adventures)
Death Note
Slam Dunk
Yu-Gi-Oh!
Ranma 1/2
Rave Master
Fairy Tail
Code:Breaker
Shirogane no Karasu
Double Arts
Monster Hunter Orage
Monster Soul (basically everything from Mashima Hiro xD)
666Satan
Saint Seiya
xxxHolic
Nightmare Inspector: Yumekui Kenbun
Samurai Champloo (anime came out first)
Phoenix Wright Ace Attorney (based off of the games, lmao)
all the .Hack// series lol
Shoujo manga:
Sailor Moon
Fruits Basket
Skip Beat!
Tokyo Mew Mew
Hanazakari no Kimitachi e (Hana Kimi)
Power!! (Girl Got Game)
Ouran High School Host Club
D.N. Angel
Zettai Kareshi (Absolute Boyfriend)
Zombie Loan (actually, I'm not sure whether this is shounen or shoujo)
Card Captor Sakura
Shinshi Doumei Cross (Gentlemen's Alliance Cross)
Vampire Knight
Beauty Pop
Yomogi Mochi Yaketa?
I.O.N
Kamikaze Kaitou Jeanne
Monkey High! (Don't really like it)
Zodiac P.I
Kitchen Princess
Mixed Vegetables (it's was so-so)
Super Pigs
Wedding Peach (I was YOUNG!!)
Arisa
Akagami no Shirayukihime
Beast Master (Def. a fav)
Kimi ni Todoke
Kaichou wa Maid-sama!
Mahou Sensei Negima!
Ghost Hunt
Koko ni iru yo!
Watashi ni xx Shinasai!
Full Moon wo Sagashite
Time Stranger Kyoko
Zettai Kakusei Tenshi Mistress Fortune (basically everything from Tanemura Arina too :P)
Tsubasa Chronicles
Shugo Chara!
Yamato Nadeshiko Shichi Henge
Gakuen Alice
Millennium Snow
Azumanga Daioh (funny x])
Lucky Star Pocket Travelers (I like the anime Lucky Star too :3)
Love Hina
Bokura ga ita
Suzumiya Haruhi no Yuutsu (Melancholy of Haruhi Suzumiya)
School Rumble
Hitomi- Tenkuu no Escaflowne
Kareshi Kanajo no Jijou (Kare Kano)
Mermaid Melody Pichi Pichi Pitch (LOL, well, I was young. So why not?)
Yumeiro Patissiere
Koko Debut (High School Debut)
Magic Knight Rayearth (Again... I was young :P)
I'm trying to rack my brain for more... Oh well, I'll just leave it at that...
And as for anime... well, let's say most of these manga were adapted into anime :)
[Edit: I feel like a freak for listing so many...]