You would want to apply as early as possble. I don't think it's necessary to fill out AMCAS and turn it in the day it opens in late May-June (...or whenever it is), but I had my primary app for AMCAS (MD) and AACOMAS (DO) in by the middle of July. And I had all my secondaries turned around within two weeks of receiving them so my applications at most MD and DO schools were complete by October. I thnk that's a good timeline since I got to hear from most of the schools before Christmas and also got my acceptances around then as well. On the other hand, you can apply a little late to DO schools so you have more leeway and time to get those in. You could wait until late Fall to turn those in whereas for MD schools it might be to your advantage to turn your applications in earlier. For me, it was important to submit early so that my application wouldn't get lost in the rush as time went on. I also appreciated hearing from more than half of my schools before the holidays. I have no experience with the Caribbean schools so I can't advise you on those. However, realise that regardless of when you turn them in, it is a long road and that requires a lot of patience. In some ways, I think writing all the essays and interviewing and waiting to hear any response from school is the worst thing anyone can do to you 😛
Be diligent and timely in turning in your applications. I think it will weigh better for you in the long run. Also, feel free to PM me if you have any specific questions about the application cycle - I am more than half way done through the hell they call "Applying to Medical School" so I would be more than happy to help out in any way I can. Good luck! 😀