You are correct. Examples:
To Think in Agreement:
Person A: "I really like bananas."
Person B: "Me too! So do I" (I also like bananas)
To Be:
Person A: "I am an engineer."
Person B: "Me too; so am I!" (I am also an engineer)
To Have Done:
Person A: "I climbed Mount Fuji in Japan."
Person B: "Me too! So have I!" (I also climbed that mountain)
In The Negative:
Person A: "I really do not like bananas."
Person B: "Neither do I," or "I don't either." (I also do not like bananas)
Person A: "I am not an engineer."
Person B: "Neither/nor am I!" (I am also not an engineer)
Person A: "I have not yet climbed Mount Fuji in Japan."
Person B: "Neither have I," or "I haven't either." (I also have not yet climbed that mountain)
I hope this helps!