How can I easily and individually update the Genre tag for a lot of songs?
As a quick-fix you might just add the "Genre" field to the list of displayed columns (Cmd-J or right-click any column name) and edit the value there directly (double-click on the current value):



There are also applications available who try to clean song metadata automatically (e.g. Tagalicious) but they only work for "well-known" tracks and will probably not tag the genre the way you need it.