I read that the awaited Messiah will cause to re-establish the Kingdom of God and redeem the Israelites so what are the evidences in Tanach and Talmud to support this claim ?
Here is one website's summary with textual sources (tanach, not talmud)
"The mashiach will bring about the political and spiritual redemption of the Jewish people by bringing us back to Israel and restoring Jerusalem (Isaiah 11:11-12; Jeremiah 23:8; 30:3; Hosea 3:4-5). He will establish a government in Israel that will be the center of all world government, both for Jews and gentiles (Isaiah 2:2-4; 11:10; 42:1). He will rebuild the Temple and re-establish its worship (Jeremiah 33:18). He will restore the religious court system of Israel and establish Jewish law as the law of the land (Jeremiah 33:15). "
It's the other way around - if someone does these things, then he is Messiah. רמב"ם שופטים > מלכים 8 & 9