In what ways did God reassure Moses with each of his excuses in the Bible/Torah?
The excuses are offered in chapters three and four of Exodus.
When God, speaking in the voice from the burning bush, first sends Moses to speak to Pharoah and demand that he release the Israelites, Moses replies to God, "Who am I that I should to to Pharaoh, and bring the sons of Israel out of Egypt?" (Ex. 3:11) God replies,
I will be with you; and this shall be the sign for you, that I have sent you: when you have brought forth the people out of Egypt, you shall serve God upon this mountain. (Ex. 3:12)
Moses tries to argue that the Israelites might doubt his authority to lead them.
If I come to the People of Israel and say to them, 'The God of your fathers has sent me to you,' and they ask me, 'What is his name?' what shall I say to them? (Ex. 3:13)
I AM WHO I AM. And he said, "Say this to the people of Israel, 'I AM has sent me to you...The LORD, the God of your fathers, the God of Abraham, the God of Isaac, and the God of Jacob, has sent me to you:" (Ex. 3:14-15)
Moses tries again, arguing that the Israelites will not believe what he tells them. God turns Moses's rod into a serpent when he throws it to the ground, then back to a rod when Moses picks the snake up by the tail. This demonstration, along with the affliction of Moses's hand with a skin disease and subsequent cure, was given by God to Moses so he could convince the Israelites of God's presence with him.
Finally, Moses pleads,
Oh, my Lord, I am not eloquent, either heretofore or since thou has spoken to thy servant; but I am slow of speech and of tongue. (Ex. 4:10)
Is there not Aaron, your brother, the Levite? I know that he can speak well;...you shall speak to him and put the words in his mouth; and I will be with your mouth and with his mouth, and will teach you what you shall do. He shall speak for you to the people; and he shall be a mouth for you, (Ex. 4:14-16)
With the provision of explanations of who sent him, miracles to demonstrate his authority, and a spokesperson to present the arguments with more refined speech than Moses himself possessed, Moses ends his excuses and assumes his leadership role.