General synonyms:
* Start
* Commence
* Initiate
* Launch
* Embark
* Set out
* Get going
* Kick off
* Undertake
More specific synonyms:
* For a process: Start, commence, initiate, launch
* For a journey: Embark, set out
* For a speech or presentation: Begin, commence, start
* For a task: Undertake, embark on
Informal synonyms:
* Get started
* Get going
* Hit the ground running
The best synonym for "begin" depends on the specific context of your sentence.