SO DO I
We use "so do I" when somebody tells you something (with an action verb in present tense), and the answer is true for you too. For example:
Your friend: I like chocolate.
You: So do I. (you could also say "I do too" or "me too")
SO I DO
We use "so I do" when you didn't think that you did (or had) something, but it turned out that you were wrong. For example, let's say you didn't think you had a book, but you were wrong:
You: I don't have the first Harry Potter book.
Your friend: Are you sure? What's that on your bookshelf?
You: Oh, so I do!
SO AM I
If somebody tells you something with the verb TO BE, and if the answer is the same for you, we use "so am I." For example:
Your friend: I am hungry.
You: So am I. (you could also say "I am too" or "me too")
Your friend: I am working on Friday.
You: So am I. (you could also say "I am too" or "me too")
SO I AM
We use "so I am" when the first sentence contained the verb TO BE, and you thought that something was true, but it turned out that you were wrong. For example, let's say that you thought you got to class on time, but you were actually late:
Teacher: You are late to class.
Student: No, I'm not! It's 3 p.m.
Teacher: Yes, you are! It's 3:05 p.m.
Student: [checks watch] Oh, so I am! Sorry about that.