What's a good starter anime for a 13 year old girl?

Well it really depends on what genre she likes. Personally I would tell her to start with something like Sword Art Online.

Action/Adventure/Romance

Sword Art Online- This is about a video game that traps everyone who stars playing. When you die in the game you die in real life. The only way to end the game is for one player to finish the last level. The main character is guy named kirito who was one of the best beta testers ( He tested out the game before it was released) and best player. He also later meets a girl named asuna who is also a highly skilled player in the game

Inuyasha- Its about a girl in high school that finds a well that takes her back in time. She meets a half demon named inuyasha. She learns that he was sealed to the tree for 50 years until she came and broke it. They travel together trying to search for shards of a jewel that is said to grant any wish you want. It was proven that anyone who comes in contact with the jewel will suffer a great deal of pain (a curse) The jewel waits for the one person who will make that one wish that will break the curse . 

Romance

Oruan high school host club- The main character Haruhi has a scholarship to Oruan high school which is for rich kids (really expensive ) Looking for a quiet place to study, Haruhi stumbles upon the Third Music Room, a place where the Ouran Academy Host Club, a group of six male students, gathers to entertain girls. She accidentally knocks over and breaks an antique vase valued at ¥8,000,000 (US$80,000 in the U.S. English ). To repay them she has to work there everyday after school till she pays it off. None of the boys actually know she is a girl because she cut her hair when she got gum in it and she dresses like a guy

Fruits basket- Its about a high school girl named Tohru who's mother dies. After she moves out of grandfather's house so it can be fix. She tells her grand father she has a friend to live with but she decides to live by her self in a tent so she isn't a burden to anyone. One day she was walking today a stumbles across a house which happens to be where one of her class mates Yuki Sohma and his cousin Shigure live. This family live with a curse. When ever they are hugged by a member of the opposite gender they change into the members of the Chinese zodiac. When she finds out, she promises not to tell anyone and decides to live there. Next she meets kyo who is another one of Yuki cousin and will also be staying with them. Yuki and kyo do not get along. During this series she meets the rest of the members of the Chinese Zodiac

Kaichou wa maid sama- This is about a girl named Misaki who is the student president of a school that was formely an all boys school. She cant stand any boys and tries to encourage more girls to come to her school. After school she works part time at a maid shop to help her mom who is struggling to pay the bills. She does want anyone to know about her job because it will ruin her reputation but one day the coolest guy in school Takumi finds out her big secret

Action/Adventure

Fullmetal alchemist Brotherhood -This is about two brothers Edward and Alphonse who lived with their mom and dad. One day their dad leaves for some unknown reason. Soon after their mom falls ill and dies. In an effort to try and bring her back they learn alchemy and try to perform a forbidden alchemic technique called human transmutation. This back fires on them. Due to the law of alchemy which is equivalent exchange (In order to get something, something of equal value must me lost) Alphonse lost his body and Edward lost his left leg. Edward also scarifies his right arm to rescue Alphonse's soul and seals it in a suit of armor.

Sword Art Online, Angel Beats

For shoujo I would say: Maid-sama or Wolf Girl and the Black Prince

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What are you interests? Action adventure etc. You may like a slice of life anime, such as toradora.

seven deadly sin, fate/staynight, fate/zero, yugioh, yu yu hakusho, inuyasha, inuyasha final act

Rurouni Kenshin-(samurai x), soul eater,kekkaishi, zatch bell, mar, RWBY,

full metal alchemist.

madoka.