There is a red dot on the top right of some of the home screen icons. What does that mean?

It means that the category is suspended for imminent/overdue alerting. To change that, if you are a manager, go to Stables->Preferences then select a value other than zero days for that category.

There is a red badge with a number over the date on the home screen. What does that mean?

It means that there are diary entries for today.

There is a blue badge with a + sign over the Turnout icon on the home screen. What does that mean?

It means that the the turnout plan has changed in the last two days.