The US consulate will not be able to help you very much. They might have some information about which Italian authorities can help you, but I wouldn't hold my breath. As far as travel is concerned, they mostly issue US passports to US nationals and US visas to non-US nationals.
You cannot get an Italian visa while you are in Italy, but you may be able to apply for a residence permit. According to the Polizia di Stato, you can apply for a permit for certain purposes, including for study, at one of the following agencies:
- Municipalities ('Comuni') which provide this service;
- 'Patronati' (institutions offering advice and social assistance to workers);
- Immigration Desk (Sportello Unico per l'Immigrazione) set up in every Italian province at the Prefettura;
- Post Offices, where you receive the kit containing the application form.
Because you are a non-EU national, you will need the kit with the yellow stripe.
Does your language school have an international student office? You will presumably need some sort of evidence from them that you're studying there, so they may have a person who deals with that sort of thing, who should also know more about the procedure.