You can use start of usually in the following situations:
If a person starts to move in a given direction, we can say "He started off towards.."
If a person or people start a project with a number of tasks, we usually say that they started off with...
"We started off by laying out the design drawings."
Some event or day that involves a number of different small events that follow or that may follow, we can say that it started off.
"The day started off normally, but..."
For any simple even or some machine, we usually just use 'started'.
"The motor was started."
"The play started at 8:00."