Of all four, Commencer is the most appropriate and Lancer the least.
Lancer means to throw or to launch and it borrows from the context of rocket launches. Je vais lancer un projet is therefore a colloquialism which should not be used in formal communication.
Démarrer would meant concentrating on the initial phase of the effort as in basically you would gather people, provide them with resources to accomplish the project (for instance funds) and then leave them alone.
Débuter is similar but is actually less and less in use. Initier is also a similar term and would probably be preferable as débuter is itself old-fashioned and démarrer is more likely associated to car use (as in "démarrer une voiture").
I would use one of the sentences below:
- Je vais initier un projet - You are the project initiator
- Je vais démarrer un projet - You are the project leader
- Je vais commencer un projet - You are likely to be alone and do it all by yourself