What's the infinitive of the verb when I say "I should go," or "I shall go"? Is it a verb actually, now that I think about it? If not, what is it?
"Shall" and "Should" are modal verbs and don't have infinitive forms.
Notice that "to should" and "to shall" don't exist.
The only verb in your sentences that has an infinitive form is "go", for which the infinitive is "go".
That's because your sentence follows the typical structure of a sentence with modal verbs:
[subject] [modal verb] [infinitive of main verb]