Manicmonkie
I remember asking a similar question about 'aerospace engineering' when I was in high school. My friends older brother, a manufacturing engineer for Magna said "What is aerospace engineering?" It was a rhetorical question, he went on to explain: "Aerospace engineering will have mechanical engineering, electrical engineering, engineering physics, chemical engineering, materials engineering etc."
So consider bicycles as you see them today. What types of science and engineering are required to produce a bicycle? What parts of the bicycle interest you most, is it the drive train components, the easily customizable components, or is it the integration of all of those around a frame that best suits the needs of the rider and the application?
Assuming your interest is in the integration and frame design area I would agree with ACtrafficengr's suggestion: Mechanical Engineering. But what things specifically will you need? Mechanical engineering covers a lot of ground: strength of materials aka mechanics of deformable solids, structures, basic physics, statics, dynamics, heat transfer, fluids, fluid dynamics, turbo-machinery, vibrations, math math and more math, then just to be sure a few more math courses, basic electrical engineering. You could apply fluid dynamics to wind tunnel optimization of bicycle and rider for time trial work, or you could master materials and earn a welding specialization and determine the best way to weld frames to ensure structural reliability and geometrical accuracy of the final weldment. The list goes on and on.
Or you could get a chemical engineering degree (aka Fem-Eng since at least at my university, so many women enrolled in the program) and become an expert at composite materials to make bikes just as strong but so much lighter.
In the end you've got to like the branches of science you apply to engineering purposes, from there keep in mind the ways these sciences could be applied to manufacture bicycles... Manufacture there's a whole area to itself!
Look up university websites, find their curriculums for their mechanical and chemical engineering programs. Read all the course descriptions for each year in the program.
Here's McMaster Ontario Canada's course index for Mechanical Engineering:
Also if you really want to go all out, spend time reading at the university web sites about the engineering professors in the mechanical engineering department, see which of them are leading research into areas you think could be applied to bicycles.. then if you play your cards right, you can apply to that school, do your 4 year undergrad in that department, and if you do well, apply to do a masters and really become an expert at some aspect of science and engineering that could be applied to bikes.
This of course is only one approach, there are many ways one could go about doing it.
Good luck! :-D