Hello!
"Αρχίζω να πεινάω" is the correct form. (να + Present Tense)
να + Present Tense is used to show a prolonged or repeated action or an action in progress.
Example:
Μου αρέσει να διαβάζω εφημερίδα κάθε μέρα. = I like to read a newspaper every day. -> repeated action
The verb αρχίζω is always combined with this form and never with the future, since the verb after it indicates an action in progress.
Examples:
αρχίζω να καταλαβαίνω = I start to understand (understanding is an action in progress)
αρχίζω να κλαίω = I start crying
αρχίζω να πεινάω = I start being hungry
να + Future Form is used to show an instant action.
Example:
Θέλω να φύγω. = I want to leave.
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Vasiliki Baskou, Instructor/Director, https://learn-greek-online.com.