Learn Hardcore Italian: Vincere una partita è importante per il bambino. - Winning a game is important to the child.

In Italian the infinitive can act as a noun and serve as the subject of a sentence. Here vincere (“winning” or “to win”) names the action itself. English often uses a gerund ("Winning a game is important"), but Italian simply uses the bare infinitive.

Could I rephrase this sentence with a subordinate clause instead of the infinitive?

Yes. You can say:
È importante che il bambino vinca una partita.
Note the use of the subjunctive vinca because subordinate clauses introduced by che and expressing importance require the subjunctive mood.

Why is una partita used rather than la partita?

Using the indefinite article una generalizes “a game” (any match). If you refer to one specific game already known to speaker and listener, you would use la partita (“the game”).

What exactly does per mean in per il bambino? Could I use a or al instead?

Per expresses purpose or beneficiary (“for the child”).

Per il bambino = “for the child’s benefit.”
If you said È importante al bambino, it would mean “it is important to the child” (the child cares about it), shifting focus to the child’s feelings rather than the outcome benefiting him.

Can I change the word order to start with È importante?

Yes. You could say:
È importante per il bambino vincere una partita.
Both orders are correct. Leading with vincere emphasizes the action; leading with è importante emphasizes the quality of importance first.

Why is there a definite article before bambino? How would I say “for a child” in general?

Il bambino means “the child,” a specific one known in context. To express “for a child” in general, use the indefinite article:
per un bambino.

Why is bambino masculine here? What if I want to refer to a girl?

Bambino is masculine and refers to a boy. For a girl, you change to the feminine form:
Vincere una partita è importante per la bambina.

The adjective importante ends in -e. Does it change with gender or number?

Italian adjectives ending in –e have the same form for masculine and feminine singular. For plural (both genders) they change to –i. So:
Singular: importante (m./f.)
Plural: importanti (m./f.)

Why is the verb essere in the third person singular form è rather than plural?

The verb agrees with its subject. Here the subject is the infinitive phrase vincere una partita, which is treated as a singular concept. Therefore you use singular è.

Is using the infinitive as a subject common in Italian, or only in certain cases?

It’s very common whenever you want to speak about an action in general:
• Mangiare sano fa bene. (Eating healthily is good for you.)
• Viaggiare all’estero arricchisce. (Traveling abroad enriches you.)
This construction is especially handy to avoid more complex subordinate clauses with che + subjunctive.