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I’ve been trying to remember a show for a while and I just can’t think of the name. Unfortunately I don't remember enough to find it through Google searches, so hopefully someone knows which show I'm thinking of.

I remember the main character is a young blond boy who goes to school in, if I remember correctly, an European country. He finds a stone(?) that his father hid in their house that can summon creatures/monsters. There are people who want the stone and use other stones to summon creatures to attack him.

He ends up joining with a girl, that is his age and if I remember correctly, goes to the same school, and a man who teaches him. The man has a stone that can summon a creature named Caliban. That’s all I can remember, other than that, I think there might have another female character with long black hair who I think might have been bad then switched sides and her name I think started with a Z.

Does anyone happen to know what cartoon I'm thinking of?

Probably Huntik, an American-Italian cartoon aired between 2009 and 2012.

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  I remember the main character is a young blond boy who goes to school in, if I remember correctly, a European country.

  
  

The blonde guy, Lok, goes to school in Italy.

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  He finds a stone(?) that his father hid in their house that can summon creatures/monsters.

  
  

He's in his house with his friend Sophie (the girl on the left), who breaks a jar, which contained Lok's father's amulet and diary. You can watch that scene in the following video:

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  He ends up joining with a girl, that is his age and if I remember correctly, goes to the same school, and a man who teaches him. The man has a stone that can summon a creature named Caliban.

  
  

Well technically he knew Sophie, his classmate, a little bit before that. He finds Dante Vale (the adult man in the picture above), who is a Seeker. Seekers can summon creatures known as the Titans, and Dante particularly can summon Caliban, as he does later in the same episode (see video below). Dante being an experienced Seeker, he has the "guide" role for the two teenagers.

Here's a picture of Caliban, in case the previous video does not play:

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  There are people who want the stone and use other stones to summon creatures to attack him.

  
  

Throughout the series, the kids fight against the evil Organization, whose goal is to rule over the world (such originality !). The Organization seeks a huge, legendary-like Titan to ensure their domination. So basically, they're like villainous Pokémon teams, but without the catchy name.

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  I think there might have another female character with long black hair who I think might have been bad then switched sides and her name I think started with a Z.

  
  

The dark-haired girl on the right (see above still) is named Zhalia. She was raised by a member of the Organization and acted as a spy for them, even betraying the crew she had come to befriend, but later rejoined them in episode 20, The Unseen Guide.

As to how I found that, my browser history displays the following results (and a typo):

I remember that at the time I first wrote this answer, the tv show "caliban" summon query returned the cartoon's opening on YouTube. As of now, the first page of results for this same query return a link to the show's wikia, a TVTrope link and this very post.

Omg my son just gave me the same description of the show that he is trying to figure out the name to! Now it is going to bother me!