“I was thinking,” she said, “that maybe you could teach me.” - (here after "said" if you put comma, in double quotation, the first letter should not be capitalized )
“I was thinking,” she said. “That maybe you could teach me.” - (here after "said" if you put period, in double quotation, the first letter should be capitalized )
If the dialog tag is in the middle of a character speaking, then the dialog is not capitalized after the tag unless it starts a new sentence. For example:
“I was thinking,” she said, “that maybe you could teach me.” [RIGHT]
“I was thinking,” she said, “That maybe you could teach me.” [WRONG]
“I love that dog,” she said. “He’s so cute.” [RIGHT]
“I love that dog,” she said, “he’s so cute.” [WRONG]