Degrees
Design Engineering: This position is dominated by Electrical Engineering (EE) degrees. Because the design engineer is often creating electrical circuits from scratch, a knowledge of electronics and how to design them is required.
Analog engineers are almost always EEs but you will sometimes find physics degrees, especially if the analog design being done is cutting-edge and requires an in-depth knowledge of device physics. Due to the complexity of advanced analog circuit design, especially in high-speed areas, a masters-level (MS) or even doctorate (PhD) degree is highly desireable. The masters, typically a MSEE, is a six year school program, whereas the doctorate can take significantly longer, but should probably be considered an eight year program for a dedicated student.
Digital design engineers have a bit more flexibility and might possess EE, computer science, computer engineering, or even software engineering degrees. Because nearly all of today's digital design processes are based upon some sort of circuit-description language (Verilog or VHDL), it is entirely possible to find even non-engineering degrees doing the job if the person has demonstrated strong ability and was able to "get their foot in the door" somehow. This is not to imply that digital design is less desireable in any way, but it's definitely less of a "black art" than advanced analog design is and generally will require a broader range of skills fillable by a wider range of degree programs.
Technician: The common technician roles of layout tech or lab tech are typically able to be filled with two-year technical degrees, making these roles a good entry position into a company. One common benefit of being a technician is that you can get your two-year degree and get a job in a semiconductor design company who will then pay for you to continue on with a full EE or MSEE degree. Because the company typically pays for the degree, it's a great way to graduate with very little debt and also to have a solid reputation within the company which will allow you to apply for engineering positions once you graduate.
Marketing: The marketing roles involved in semiconductor design, compared to general business marketing, are much more technical. Many people in the technical marketing field have some type of technical degree, such as electrical engineering, but then also have a business degree (minor or major) or perhaps even an MBA on top of that. This allows them to communicate effectively with the customer during the technical requirements definition stage, as they can discuss the "engineering stuff" but their business education allows them to understand the concepts of pricing, negotiation, and other non-technical aspects. Because a good technical background is so desireable in these roles, you will often find people who were once engineers but who decided they wanted to do something different, so they went back to school and obtained a business degree and moved into technical marketing.