As you know SharePoint calendars are views of a list type containing columns including date and time.
You can add a choice category column of your choosing to your list and by using multiple views (overlays) use this to colour code entries in a calendar view.
The example below uses a filter called Task Status with options New, Started, Complete to be displayed with separate colours.
Category name and filter options are of course customisable.
Create your calendar in the usual way
Browse to SharePoint site
Select Add an App
Choose Calendar, name it and click Create
Add Category Column item
Browse to the new calendar
From the menu ribbon select CALENDAR and click List Settings button
Click Create column
Name as Task Status, select type as Choice (menu to choose from), enter choices New, Started, Complete and select Drop-Down Menu or Radio buttons (you don’t want to allow multiple choices, so don’t use Checkboxes), and click OK
Create Calendar views
Return to the Calendar view, and from the menu ribbon select CALENDAR and click Create view button
Select Calendar View, name New, add a Filter option selecting Task Status and is equal to New and click OK
Repeat and create additional views named and filtered for Started and Complete
Create a new view called All Tasks, set to it Default View and use is equal to filter against Task Status but leave filter option blank. (This creates a ‘clear’ default view – this is to replace the default ‘Calendar view’ which would show duplicate values if used)
Create Calendar Overlay Views
From the menu ribbon select CALENDAR and click Calendar Overlays button
Click New Calendar
Add name New, select required colour from the Color drop-down
Click Resolve new to the Web URL field to load the site and confirm (or select) the appropriate Calendar in the List menu
Select New from List View, click Always Show and click OK
Repeat for Started and Complete then click OK to save the Calendar Overlays
The calendar default view will now display the three views overlaid showing each category as a different colour.