Becoming an Engineer in the United States
In the United States, the standard engineering degree is a Bachelor of Science. Billed as a four year degree, it frequently requires five years to complete. It does not normally include a thesis, but students in their final year often complete a one or two-semester group design project.
High school students who want to major in engineering should, of course, focus on math and science (especially physics) courses. Calculus and calculus-based physics would be great, but they aren't strictly necessary; almost all students take these in college regardless of their high school background. Scoring high on the ACT and SAT standardized tests is important for getting scholarships or admission to one of the more competitive schools. Engineering colleges often boast the highest average scores in their universities.
At many schools, the first year of study is the same for all engineering majors: calculus, computer programming, chemistry, physics, english, and engineering graphics (a colossal waste of time for the EE majors).
Bifurcation into the separate engineering fields begins in the sophomore (second) year. All students take more math (differential equations and possibly linear algebra) and their second semester of physics if they haven't gotten it already. Most take basic mechanics courses (statics and dynamics), the exception being students in a degree program that doesn't expect professional licensure later on, like some computer (but not electrical) engineering programs. They also take the beginning courses in their discipline: circuits and digital logic for EEs, mechanics of materials for MEs, and so on.
Most dropouts occur in the first two years. If you can make it this far--and you still want to--you can make it the rest of the way.
By the junior (third) year, most of the overlap between disciplines has ended as students take more advanced courses in their majors. EEs like me take courses in electronics, electromagnetic theory, and electric machines. MEs take advanced mechanics classes, ChemEs take thermodynamics, etc. Most students take statistics and technical writing classes at about this point.
In the final year (or years), students begin to specialize within their disciplines, taking advanced elective courses. Most schools require a design class during which students work in small groups on fairly extensive projects.
Prior or coincident to graduation, students are encouraged to take the Fundamentals of Engineering (FE) exam, one of the prerequisites to professional licensure. Being licensed as a Professional Engineer is very important for civil engineers, but somewhat less so for other kinds; if you're going into VLSI design, you probably needn't bother.