Taking Lufthansa from NYC to Munich and then, on the same day, carrying forward to Italy. To which consulate should I apply for my Schengen Visa?
It depends on what you plan to do (the main purpose of your visit) as Willeke mentioned. Say if you have a business visa and have a meeting, then you would apply to the consulate where you have the meeting.
Whereas, if you have a tourist visa and your main tourism will occur in Italy you will apply at the Italy consulate.
The Visa should be in your case from Italy , since you are transitting from Munich.
If however you intend to break your journey even for a night in Munich , you should go for Germany.
Schengen Visa Rules for first port of entry
It is the country that is your first destination in the region. More often than not, you will have to apply to the country which is your port of entry. As per the Schengen visa rules first port of entry is usually the country that issues your visa.
Check with the airlines where the immigration is going to happen , most likely it will be Italy.