Starting:
* Commence: More formal than "begin"
* Initiate: To start something formally or officially
* Launch: To start something with energy or excitement
* Embark: To start a journey or an important undertaking
* Inaugurate: To officially start something, especially a new era or project
* Kick off: Informal, to start something with enthusiasm
* Set out: To start a journey or a task
* Undertake: To start a difficult or important task
Continuing:
* Resume: To start again after an interruption
* Continue: To keep doing something after a pause
* Proceed: To move forward, to continue
* Carry on: Informal, to continue doing something
Specific contexts:
* Start: (More general than "begin")
* Open: (For events, activities, or businesses)
* Enter: (For a state or situation)
* Break ground: (For construction projects)
The best word to use will depend on the specific context and the desired tone.