(1) Work on a project (or several mini-projects) that is (are) assigned by your supervisor. This may be a follow-up project of another colleague, or a project with very slow progress because the person who's working on it has other committment.
Pros: You don't have to start from scratch; perhaps get 1 or 2 collaborative papers (of course your won't be the first author!)
Cons: The project may not be your real interest, and you have "wasted" your valuable time to be the labour of your supervisor.
(2) Work on your own idea (if you have any in your 1st year or before PhD).
Pros: Since it's your idea, you'll be more committed and dedicated to the project; get first-author good paper if you're lucky enough;
Cons: May get less support from peers if you're working on topics "outside" the expertise of the lab; more risky in terms of time...
How do you begin your PhD project?
I think PhD students in a bigger lab (like where I'm now, with 20+ people, mostly post-docs) tend to work closely with a small group of people, rather than working on his/her own project.
Do you think it's true?