First of all, glad to join the community here.
I have found that the best way to get started in a Mechanical job in the oil and gas industry known to be one of the largest employers of Mechanical Prod, is to begin by having a look at an oil & gas specific job board and seeing which roles are available. In my opinion, it's important to use a job board as you maximize the number of jobs in one place, and avoid missing any great opportunities...saves you time endlessly scanning the internet for the best jobs! The oil & gas job board at the moment with the most jobs listed is http://www.oilandgasbestjobs.com/ so it may be worth giving that a try as a starting point. Once you have a good idea of the type of jobs which are available for your skills, tailor your resume to emphasize the points which would be relevant. Then simply start sending your resume off for the jobs you like the look of (you just need to register/upload your CV once and you can apply for as many jobs as you like).
Here is an example of Senior Mechanical Tank Engineer (LNG / Cryogenic) Job.
I hope this helps!