So what anime should I start with?

[Beni]

There is this one place I heard of called something like The Cat Cafe that sounded really nice.

I did not try the cat cafes but a friend tried it and liked it very much. She bought a cat afterwards and did not return to the cafe ever again

I went to Gundam Cafe, it was interesting but I do not know Gundam series. So I did not enjoy it as much as a fan of Gundam would.

Sometimes I go with friends to maid cafes, its kind of fun there with the food and photos.

[Ghost]

Sounds very old school. I may try it when I get time to go through it.

Well, there's only five episodes out so far, out of a planned... seven, I think? They make like two a year or something.

[Hardcase678]

Iria Zeiram The Animation and Akira.

[CoconutComa]

If you're looking for something like harem, romance, tragedy, take a look at Clannad and other related works such as Kanon and Air. These are pretty good starters.
If you're going towards the sci-fi, I definitely recommend Steins;Gate. It's 24 episode series that revolves around a group of friends who perform experiments that are motivated by the notion of time traveling. With this said, it ties in the theme of cause and effect which drastically impacts the characters development. Steins;Gate moves at a good pace with bits of comedy, suspense, and plot twists.

[Kage Kisaragi]

Hello Kitty.. :)

[Ambient_Malice]

Kino's Journey. Only 12 or so episodes, but amazing, mature stories with a strong female lead. The art style isn't typical, and sorta resembles Astro Boy in some ways, but the anime is anything but kiddie. I heartily recommend it.

[ItsFantastic]

Try Sword Art Online.