I wanna start watching Anime... Where should I start?
There's way too many options to start with.
Pushing the long list below aside, right off the top of my head I'll suggest Tenchi Muyo, Outlaw Star or Gundam Wing. They're not the first Anime I seen, but they're the ones that cemented my fascination with Anime.
If you want to start somewhere like I did, you're gonna have to go old school. Sometime around the 80s to early-mid 90s Anime. All Purpose Cultural Cat Girl Nuku Nuku, Dirty Pair Flash, Bubblegum Crisis Tokyo 2040 (which Wikipedia is saying was released in 1998-'99... BULLSHIT) just to name a few.
Now, if you're going for anime you can just find on the TV, your choices are fairly limited. Independent Film Channel has Hell Girl, Sci-fi's current Anime series is Gundam 00. The rest are on Cartoon Network. When it comes to CN, the only time they show the good anime these days are very late at night, and only on Saturdays. So if you're lucky, you can catch Moribito, Full Metal Alchemist or Cowboy Bebop on that night.
If you want to start with Anime Movies, there's actually a good number of them out there. If you find any of Hayao Miyazaki's work, GET IT. Porco Rosso, Spirited Away, Nausica of the Valley of the Wind, Princess Mononoke, Laputa: Castle in the Sky, etc. (too many to count.) However, when it comes to Nausica, be very careful not to get the Wind Warriors, that's a heavily butchered version of the movie that almost made sure no other Miyazaki film would touch foreign soil. Akira is another great movie as well as Steamboy. You can also try the Appleseed movies which if my math is right, there are three.