For me, when working on all encompassing projects that take up all of my time, I move through phases of what I do over a 6 month to 2 year cycle. We start with requirements and concepts, move to detailed design and costing, then it is on to prototyping to proof of concept, then testing. And you may have guessed it, we take the data and circle back to detailed design and verified we fixed the issues. Eventually we move to tool release to make production parts, and we are back at testing and debug. Then we do pilot builds and test. And back again to testing and debug. Eventually you hit production release. More issues show up when you hit high volume ramp up that also need to be resolved. Depending on the company size and organization, you may own the product until end of production.
Short answer is that every day is a bit different and will change project to project.
Being an engineer involves balancing many requirements for a project. As a mechanical engineer, you may have to design a part that goes into a larger system. There will be many factors to that part:
What material to use?
How precisely does the part need to be?
How expensive can the part be?
What materials can it be made out of?
What is the best shape?
What forces does it need to resist, and for how long?
This is only a small list, and being an engineer is finding the best place on the sliding scale of each of those factors for the thing you're designing. It is a long process that you will usually be working closely with other engineers from similar and different disciplines.
Design integrated mechanical or alternative systems, such as mechanical cooling systems with natural ventilation systems, to improve energy efficiency.
Research, design, evaluate, install, operate, or maintain mechanical products, equipment, systems or processes to meet requirements.
Read and interpret blueprints, technical drawings, schematics, or computer-generated reports.
Specify system components or direct modification of products to ensure conformance with engineering design, performance specifications, or environmental regulations.
Recommend the use of utility or energy services that minimize carbon footprints.