See, that's the thing. Your degree is just an indication that you understand the principals of engineering and can apply them to some degree. What you actually do in the field doesn't necessarily have to be what you got your degree in. For instance, if you get your degree in Aeronautical engineering, you may find that you are doing "mechanical" work for your entire career. Conversely, you might wind up doing the strict aero stuff. It really depends on the company, the industry, your goals, etc.
Aero engineers do a lot of different things, but generally they are responsible for, as you might imagine, aerodynamics. They may work on control surfaces, or performance characteristics of various geometries (like delta vs straight airfoil planforms, stuff like that). I do not work as an aero engineer in industry, but the main thing you should realize is that something like "aeronautical engineer" aren't really job titles/descriptions. That is a broad range of job descriptions. Aero's definitely fit into the engine design "category", since they must be familiar with the principals of propulsion and are generally versed in nozzles and supersonic flows and that jazz, so I imagine they'd be part of it, but most of the work is likely to be what you might consider a "mechanical" engineer's job. The space industry is one of those that tend to have fairly specific types of engineers. Whereas the mining industry might have "Structural, Electrical, Piping, Mechanical" as major groups, the space industry has loads of 'em, "Propulsion systems, control surfaces, cooling, system control, electrical, bla bla bla".
Also note that, if you want to do R&D, an undergraduate degree will likely not be the end of your formal education. In general, research is carried out by folks with at least a masters in their chosen field, and in many/most instances, a PhD. That is not to say that you can't work on propulsion systems and do just as interesting work, but there is a huge difference between altering and improving on current designs, and coming up with new ones altogether.
Again, though, understand that in industry, you are not as limited by your degree as you might think.
