He  Don Vito nodded and chimed in. “Yes, of course. Sicily is the southern center for regain of art. And by the way, you can stay by with us. We have a fantastic villa. from it where you can see the entire Valle dei Templi.” 
“Really? The Valle dei Templi is one of my dream places to go!” Monica confessed.

“And our villa overlaps in part of the Necropolis of Pantalica…”
“The Necropolis?” Monica was enthusiastic. 
Don Vito nodded again. “So, will you be Our guest then in Sicily?” The conversation should be friendly and inviting, so shouldn't be so formal.  Of course the whole point of the conversation is to stay in Sicily and it would be silly to be so explicit again.
Monica looked at Alexander for a moment. She had promised him to spend the summer camping together…
“Accept the invitation or no Art Academy at all.” Her mother declared triumphantly insisted friendly.

Remember your characters!  The language used by each is different.  For example, Mrs. Deluca's words are proper and carefully thought out before utterance as she is not only imperious but calculating.

“My daughter is the top student of her class!" a high school graduate now and with the best marks.” Silvio De Luca tried to divert  steer to another topicthe conversation to another subject. “And soon She’ll be the make a great general manager of in my telecommunication empire! Won’t you, my dear? ” He said proudly, clutching Monica by her rested his hands on her shoulders.
Monica nodded with a forced smile. “Daddy... In reality,  pa, I love to drawing so much that I’d like to attend the Art Academy in Venice…”

“What?” Mrs. De Luca was incredulous upon hearing her words. couldn’t believe her ears. Her daughter in an academy full of filled with silly, young people nit-wits with frivolous a lot of dreams and no money nary a penny in their pockets? 
“Don’t worry mam. I’ve just found a nice flat with good girls roommates. and I’ll do all my exams on time without losing will study hard and not to fritter my time away. I promise!” Monica crossed her fingers to prove her earnest seal her oath. “Mam, let me go and I’ll do everything you ask of me. Please!?”

A trip to Sicilian Trip

Monica smiled blushed meekly in smiles even though a little bit embarrassed. her attention was concentrated on the poor Alexander, who was picking up all the glasses shards with his Aunt Li.

A trip to Sicily

Silvio De Luca tried to divert the conversation to another subject. “And soon she’ll be the general manager of my telecommunication empire, won’t you?” He said proudly, clutching Monica by her shoulders.
She nodded a forced smile. “In reality pa Actually, I love to draw so much that I’d like to attend the Art Academy in Venice…” (if it is a young person speaking they would use the word 'Actually' or 'Sorry Pa but'..')

“What?” Mrs. De Luca couldn’t believe her ears. Her daughter in a Academy full of silly, young people with a lot of dreams and no money in their pockets.

“Don’t worry mam. I’ve just found a nice flat with good girls and I’ll do all my exams on time without losing time. I promise!” Monica crossed her fingers to seal her oath. “Mam, let me go and I’ll do everything you ask me, please!?” ( or 'You asked of me')
Mrs. De Luca’s eyes glittered for a moment as she saw the solution to her diabolic plan. “Okay Sweetie, you can go to the Art Academy…” Monica hugged her mother so hard she almost suffocated her. “ But… only after a nice trip to Sicily to deepen you artistic and cultural preparation” Mrs. De Luca said categorically and gave a look to Don Vito.
He nodded. “Yes, of course. Sicily is the regain  region of art. And you can stay by  near us. We have a fantastic villa. From it you can see the entire Valle dei Templi.”
“Really? The Valle dei Templi is one of my dreams !” Monica confessed.( 'one of' implies there is a group of things to choose from and you picked one so the group name has to be plural)
“And our villa overlaps  part of the necropolis…”
“The necropolis?” Monica was enthusiastic.
“Don Vito nodded again. “So, will you be our guest in Sicily?”
Monica looked at Alexander for a moment. She had promised him to spend the summer camping together…
“Accept or no Art Academy.” Her mother insisted friendly.( this is odd because the have slightly different connotations/ ideas. insisted sounds like her mother is forcing her whereas friendly implies there is a choice. It sounds like she does have a choice) 

Overall, this is a really good piece, I was really intrigued. The mistakes in my view are minor, more to do with the subtle meanings of the word.