Hello @Landoexist ,
The dot size is part of the symbology, and symbology is not permanent with the data, it is only applied in the map as a way of displaying the data. So it is expected that the symbology disappears and returns to its default state when you remove the data and re-add it to the map. This will be true in ArcMap, in ArcGIS Pro, with Python, or any other option.
Add your data to a map, symbolize it, and save the map. Find out from your teacher how they want you to submit the assignment. It might be a pdf of a map showing the symbols and the legend. It might be the .aprx map document file zipped up with the data. It won't be just the data itself, because the data alone cannot store the symbology and legend.
You can symbolize your data by size in ArcMap usiong either Graduated symbols or Proportional symbols. Here is an example of using Graduated symbols in a lesson: https://learn.arcgis.com/en/projects/get-started-with-arcmap/#subsection-3
Extra information: You may also be interested in saving a layer file. This file will store the layer properties for a dataset, including its symbology.