The project for my class involves downloading a zip (with sharefile documents) + csv file which provide the data and a basic map of Manhattan. As per instructions, I computed a new field and created a map based on it. Whenever I try to save this file as a shapefile, it only shows the information from the original zip file (and none of my work).
Hopefully this makes sense, I'm new to GIS.
In Qgis you save your work into project files which have the file extension .qgs. You do not save your work into a shapefile.
On the file menu if you do Project/Save or Project/Save As you will be able to save your work with a name you choose. If you call the project "homework" the file saved will be homework.qgis.
The project file knows which shapefiles or other data you used for the project and remembers where that data is stored.
I believe that QGIS can't save changes back into the zip. Unzip first and open the uncompressed shapefile for editing. Zip the edited shapefile into a new archive for your teacher. Some teachers do also guide their students if they have problems.