“So am I” or “So do I”?

You use “so am I” when you are responding to some form of the verb be:

I’m hungry. So am I.
I’m eating ice cream. So am I.

You use “so have I” when you are responding to some form of the verb have. You will always use “so have I” when have is the auxiliary verb for the present perfect:

I’ve eaten too much ice cream. So have I.

You may use either “so do I” or “so have I” when have is used on its own to denote possession of something:

I have an ice cream cone. So do I/so have I.

Americans in this case will pretty much always use “so do I”, whereas British will use one or the other.

Other auxiliary verbs repeat the verb used as well:

I would like some ice cream. So would I.
I will bring some ice cream. So will I.
I should eat some ice cream. So should I.
I must eat some ice cream. So must I.
I ought to eat some ice cream. Me too!

(“So ought I” sounds hopelessly stilted. You can always say “me too” instead of any of these.)

Ordinary verbs will always use “so do I”:

I like ice cream. So do I.
I need to eat some ice cream. So do I.