@Ltryptophan Mechanical engineering core subjects are engineering mechanics, thermodynamics and heat transfer, fluid mechanics, machine design,...and machine design is one of it. I will try to explain it in short.
When a company creates a new product, it starts with market research to know what customers want (comes under marketing). Using these customer requirements, R & D people create a concept (and make and test a prototype). Then, it is job of designers to get this concept from R & D and make it understandable to people who manufacture it. Culmination of design work is the drawing. This drawing contains dimensions, tolerances, specifications (material or any other like surface finish, hardness, etc.). Designers are responsible for fine tuning shape to balance weight/moment of inertia, reduce stresses (physical/thermal) and to ensure it fulfils the requirements with safety considerations. So, that would require mechanics knowledge – stress, strain, etc. in addition to at least basic knowledge of thermodynamics, heat transfer. Maths would come into play here. Depending upon what product/industry, you would require different maths. For example, mould design for injection moulding of plastics, knowledge of Laplace transform will be required. In contrast, for people in manufacturing (almost) no much engineering maths is required. After manufacturing, it goes through quality, dispatch, sales and distribution, and in-between storage.
You may need to decide specifically which design that interests you. Product designers usually do things that I explained in the above paragraph. Of course, they now use computer packages. There are process designers too. They will possibly decide about which manufacturing processes will be used, what kind of machinery is required, what tools and accessories be needed, etc. Industrial engineers design the work-space to make it more productive. Then there is industrial design if you are artistic; probably you wouldn’t need maths here, but you should be good at drawing/sketching. There are typically car designers. You may take a look at following links.
Mechanical engineering design a good book by Shigley.
Industrial design school ranking in US
…from around the world.