Performing an Engineering Analysis

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I would like to learn how to perform a detailed engineering analysis on a product/machine but I am unsure on how to go about it. I know that an analysis involves doing a literature review by reading journals, papers about the device, doing some stress analysis tests, CFD on the fluids it utilizes that sort of thing. But what I do not know is what kind of methodology would I have to use? Where do I start? What kind of step-by-step procedure do I follow if my end goal is to write a report showing that I have done the analysis and to recommend improvements?
 
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That all depends on what the machine does.

Is it a transportation device, e.g. car, bus, truck, plane, helicopter, ship. Each has different functions, and are made of different materials.

One could focus in the propulsion system, or transmission, or structure.

Stress analysis is one aspect. Corrosion and wear and tear (tribology) might be another.

If the machine is used in a manufacturing process, then down time and damage to a product are considerations, as is throughput and energy consumption.

Or power systems. One might do a detailed thermo-mechanical analysis looking a life cycle issues, energy consumption and efficiency, corrosion studies, pollution study, cost, decomissioning and disposal study. All of these would apply to a nuclear power system.

Sophisticated tools like CFD may not be necessary, but these days, perhaps they are more than likely part of a systems analysis to improve or optimize performance.
 
"Engineering analysis" is a pretty vague descriptor. It can have so many meanings.

Like Astro pointed out, what ever you are doing the analysis on will drive what is included in it. Another wrinkle will be that the analysis could be mandated by someone with a very specific scope. It may only concentrate on a very specific part of the whole picture.

Do you have a specific task in mind? It may help narrow down all of the possibilities that your analysis can contain.