Undertaking a mapping project, whether large or small, usually starts with the question do I have source material with locational aspects? If the answer is yes, the next steps can look something like this:
Identify your locational data. You may already have a list of locations with some information about each. Or, you might be dealing with prose from a historical source that mentions several places.
Structure your data in spreadsheet form. Each row equals one geographic point. Mapping software needs at least a few separate columns for the location information itself (i.e. city | state | country). If you need to be more precise you can include a street address or even the exact latitude and longitude of your point (again, each in their own column). Beyond this sort of data, you can add any number of additional columns depending on the material you're working with—at minimum a title/label but perhaps also description, quantitative figures, and dates. Be consistent with how you enter different categories of data.
Choose your GIS tool. There is a wide array of desktop and web-based GIS tools, and they vary widely in terms of function and learning curve. For a quick look at your dataset, you could use a simpler web-based option such as ArcGIS Online or Carto. For more advanced kinds of visualization and analysis, you may want to consider the desktop version of ArcGIS or its open source analog, QGIS.
Load your spreadsheet into a GIS tool. Most common GIS tools have a function to automatically put your points on the map based on the locational data you provide (this process is called geocoding). If the tools gives you the option, make sure that it is looking at the correct columns in your data for the locational information.
Backup your data. By now you should already have a spreadsheet version of your data. Once you load it into a GIS tool, you should be able to export that data either as a spreadsheet (CSV) with latitude and longitude columns or as a GeoJSON file (both are recommended). These formats are text-based, which means you'll always have access to your data, even if particular GIS tools change or go away. Save versions of your dataset as you edit and add to them.
Share your map. The easiest way to share the maps you make is through one of the web-based mapping tools.