With all one's strength
The Talmud (Shabbat 119b) teaches: "Whoever answers Amen Yehe Shemeh Rabba with all his strength — his harsh decree is torn up." Answering with intent is the heart of Kaddish.
Until "Yitbarach"
The Beit Yosef wrote not to pause between "Almaya" and "Yitbarach", and Rav Ovadia Yosef wrote (Yabia Omer vol. 1, 5) that Sephardim answer until "Yitbarach".
In the synagogue
Listening, staying silent and answering aloud are all part of honoring prayer. This article is for learning, not a halachic ruling; for practical questions, ask a qualified rabbi.
Sources
שבת קיט ב — כל העונה אמן יהא שמיה רבא
Kaddish meaning · Amen Yehe Shemeh Rabba · Maran teachings