Gioco a pallone con i miei amici in giardino.

In Italian, when you talk about playing a sport or game you use giocare + the preposition a + the name of the sport/game. You never say giocare + definite article + sport. For example:
• giocare a calcio
• giocare a tennis
• giocare a pallone

Yes, but the nuance is different:
• giocare a pallone means “to play the ball game” (often soccer among kids).
• giocare con il pallone means “to play with the ball” (kicking or tossing it around), without implying a formal game.

Possessive adjectives in Italian normally require a definite article. Since amici is masculine plural, you pair it with i + miei:
• i miei amici
Without the article (con miei amici), the phrase would be ungrammatical.

Certain location expressions drop the article after a simple preposition:
• in casa, in giardino, a scuola
Here in giardino functions adverbially (“in the garden/yard”). If you use nel (= in + il), you’re specifying a particular garden (“in the garden of that house”), which is correct but more marked.

Both are possible but have different subjects:
• Gioco a pallone con i miei amici places the focus on io (“I play ball with my friends”).
• Giochiamo a pallone in giardino makes noi (“we”) the subject: “We play ball in the garden.”
Choose gioco for “I” and giochiamo for “we.”

• Giocare a pallone is a casual, general way (especially among kids) to say “play ball,” often implying soccer but in a colloquial sense.
• Giocare a calcio specifically means “to play the sport of soccer/football.” Use calcio when you want to be precise or in a more formal context.