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I am very new to GIS and ArcGIS.

I have an Excel file containing data on cities (only names) and provinces (of those cities) for 158 countries around the world. The data also contains some economic (e.g. employment level) and demographic data (e.g. population).

I would like to visually analyse (map) the data via ArcGIS.

Do I need to add some geographic data (eg. latitudes and longitudes) into my Excel spreadsheet so that ArcGIS read it and, if yes, where do I get it from?

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Cities can be reasonably approximated as points, and it is therefore possible to put columns for latitude and longitude in the Excel file and then import them into ArcMap. To find those coordinates, there are a wide array of websites; some have well-known cities in a table [example], others let you look up the city [example], and a quick Google search will turn up many alternatives.

Esri has a walkthrough of importing XY data which should enable you to get the points into a map after you've entered it into Excel.

Provinces (or countries, etc.) wouldn't really look right as points, however. For those you should first find polygons for the administrative boundaries you're interested in. (I'd actually suggest looking through data from ArcGIS Online first, since it has many commonly used boundaries and may meet your needs.)

Then you can Join the Excel data for those provinces to connect the non-spatial attributes to the polygons. This will require a common field shared by both the polygons and the Excel file -- a numerical code, or a name, something that indicates how the program will know which rows match which data. (Note that this approach can also work for your cities/points, if you're able to find a dataset of points that covers all the cities you need!)

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In your case, getting coordinates will make points that don't help to visualize the countries. I would rather recommend to download a shapefile with the administrative boundaries (e.g. on gadm), then you can use a GIS software (e.g. QGIS) to link the names of your countries/province with your table in excel

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