I was watching a class on Zaad Al-Ma'ad by Shk Abu Mussab. I saw many different Duas after before the Salam. So I was wondering do we say the different Duas in all the positions of Salah at all times? (Ex: First rakah I say اللهم ربنا لك الحمد and next rakah لِرَبِّيَ الْحَمْدُ)

No, you don't do that. That's an excellent example you've given, because that's exactly where you only choose one and say it. You say: رَبَّنَا وَلَكَ الْحَمْدُ (Rabbana wa laka al-hamd). That's it. Baarakallahu feek. As for du'a in sujood, for example, you say: "Subhanak Allahumma Rabbana wa bihamdika Allahumma-ghfir li," and then afterwards you say: "Subbuhun Quddusun Rabbul-Mala'ikati wal-Ruh." After, of course, saying "Subhana Rabbiyal A'la" at least once, then this is allowed, as al-Imam al-Nawawi (rahimahullah Ta'ala) has mentioned.