Three ways to order. One is blunt, one is polite, one is what Italians actually say.
Grammatically correct. A bit blunt. Works fine, but sounds like you’re making a demand rather than an order.
The one to use. Polite, natural, what you’ll hear at almost every table in Italy. Literally “I would like” — conditional tense, but you don’t need to think about that.
Literally “I take” or “I’ll have”. Very common, very Italian. Sounds confident and natural — like you’ve been eating in Italian restaurants all your life.
None of these is wrong. At a formal dinner, vorrei. At a trattoria with checked tablecloths, prendo fits perfectly.
You’re talking to someone for the first time. Formal register.
At the table, the waiter is already expecting you to order. Either one sounds natural.
Requesting something not on the table: water, bread, more sauce.
There’s sometimes more than one correct answer, but one option is always more natural for that situation. Pick the best one.