Oh yes, those are definitely some popular ones! I did love me some Pokemon when I was younger as well. And of course all of Hayao Miyazaki's films are wonderful (I really like Howl's Moving Castle and The Cat Returns).

If you are not really used to watching anime, the Psychic Detective show might throw you off a bit. But for a beginner, I would suggest:
-Fullmetal Alchemist: Brotherhood -> The series follows the story of two alchemist brothers, Edward and Alphonse Elric, who want to restore their bodies after a disastrous failed attempt to bring their mother back to life through alchemy. I have yet to find someone who doesn't like this show. It's the first anime I started watching that really got me into it all.
-Death Note -> It centers on Light Yagami, a high school student who discovers a supernatural notebook, the titular "Death Note". This notebook grants its user the ability to kill anyone whose name and face they know. The series follows Light's subsequent efforts to create a criminal-free utopia using the notebook, and the efforts of a detective known as L to stop him. This is less action-packed and more of a mind game, but again it's rated as one of the top anime series out there.
-Samurai Champloo -> The story is set during a fictional version of Japan in the Edo period with an anachronistic, predominantly hip-hop, setting. It follows Mugen, an impudent and freedom-loving vagrant swordsman; Jin, a cool-headed and stoical ronin; and Fuu, a brave young girl who asks them to accompany her in her quest across Japan to find the "samurai who smells of sunflowers". This anime will really help you decide if you want to delve any deeper into other series lol. I really enjoyed it, but it is definitely unique and show's how different anime is from regular cartoons.

Anyways, i'm glad to hear you want to watch more! I hope you like these things! Of course there are so many different ones out there I'm sure there is something you will really enjoy. The hard part is taking the time to find it!

Oh, and I forgot to mention that Ghost Hunt, Samurai Champloo, and most of Fullmetal Alchemist: Brotherhood are available on Netflix right now. The others you can find easily by just typing in the name of the show and then typing the episode you want (i.e. death note episode 1). Almost all anime is available online for free and this is the best way I have found to search for it.