(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?

Wow, quite a few questions in there. I don't know where to start.

Well, if you've got an idea you're passionate about and have the go ahead to do it, then go for it. You might not have the full cooperation/resources available to you but you will definitely get some assistance from peers and other academics and i'm certain that you would acheive more satisfaction in the end working on your own stuff.

Hi Yick,

my project is my own idea, and the advantage is that it is interesting yet the disadvantage is that I have to find everything out myself and the enthousiasm of my supervisor(s) is , to say the least, limited.

Maybe in their project it would be the other way round making life a lot easier!

Hii Yickkk
let me say that Im excatly in the same position as yours
for my case, my project is my idea, and no one in my lab has worked on it..nor my advisor is much aware of it..im yet to make my mark in research field with any sort of publications...(this is my 39th month of PhD)...I have to do everything by myself...there are nobody in my lab who are expertise in my area...i will be SOLELY responsible for the success and failure...
inspite of all this..i feel itz better to work on our own project...since after completion..u can proudly say that..yes it was mine..and mine only......