Dopo che ho finito la lezione, bevo acqua in giardino.

In Italian, when you link two clauses and the second clause has a finite verb, you need che after dopo:
• Correct: Dopo che ho finito la lezione,…
• Incorrect: Dopo ho finito la lezione,…
If you want to drop che, you must use a non-finite form:
• Dopo aver finito la lezione,…
• Dopo la lezione,… (using a noun phrase)

Both express “after finishing the lesson,” but:
• Dopo che ho finito… uses a finite verb (ho finito) and is very common in speech and informal writing.
• Dopo aver finito… uses an infinitive construction (aver finito) and sounds slightly more formal or literary.
Meaning-wise they’re interchangeable.

Italian often mixes a completed action with its immediate result or habitual follow-up in the present tense. Here it can mean either:
• A one-time sequence: “After I have finished the lesson, I drink water in the garden.”
• A routine: “Whenever I’ve finished the lesson, I drink water in the garden.”
If you meant two past events on the same day, you’d say: Dopo che ho finito la lezione, ho bevuto acqua in giardino.

• Bevo acqua is a general/indefinite statement: “I drink water” (in general or right now).
• Bevo dell’acqua uses the partitive (some water) and highlights “some amount of water.”
• Bevo l’acqua means “I drink the water” (a specific water, e.g. the bottle on the table).
All three are correct, but each has a slightly different nuance.

When you speak of a place where you habitually are or perform an action, Italian often omits the article:
• in giardino = “in the garden” (general location).
Adding an article (nel giardino) makes it more specific (“inside that particular, maybe enclosed garden”). Al giardino is less common for private gardens and often refers to “to the public park/garden.”

Yes. When a subordinate temporal clause (Dopo che ho finito la lezione) comes before the main clause, you separate them with a comma. If you reversed the order, the comma could be optional:
• Bevo acqua in giardino dopo che ho finito la lezione.