If I understand correctly: you have a CSV of all airports, including the lat-long coordinate. You can load this into QGIS as a "delimited text layer" and map it.
Maybe then best to "save as" the "delimited text layer" as a proper spatial data file - save it as a GeoPackage, then reload and you don't need the CSV any more.
Then you need to enable edits on it and add a new column, called something like
"visited", and set this to
True for all airports you've been to. Save your edits.
Then you use QGIS' layer styling to show visited airports with a red dot (or whatever) and non-visited airports with no symbol. You could do this by using a categorical symbol in the symbology dialog. You could also do this with a "Filter" on the layer.
Problems might occur if, for example, new airports appear and are added to the CSV from the
"ourairports" source. It might be better then for you to create a non-spatial layer of just the airport codes you've visited, and then use that to select from the layer of all airports - but this requires a bit of database manipulation knowledge. I could set this up as a QGIS project if I had a bit more time...