For general use:
* Started: This is the most common synonym and is often interchangeable with "begun."
* Commenced: This is a more formal synonym.
* Initiated: This emphasizes the beginning of a process or action.
* Launched: This suggests a more forceful or ambitious start.
For specific contexts:
* Embarked: This is used for journeys or projects.
* Undertaken: This is used for tasks or challenges.
* Set in motion: This emphasizes the start of something that will continue.
For emphasizing the past:
* Has begun: This emphasizes that the action has already started.
* Has commenced: This emphasizes that the action has started formally.
* Is underway: This emphasizes that the action is ongoing.
The best word to use will depend on the specific situation and the desired tone.