I'm assuming that you are not talking about yourself, but a fully trained scientist. If you ARE referring to yourself then you can "sign on" in high-school by doing well in your science courses. Follow this up with 4 or 5 years majoring in one of the biological sciences at university, then probably another 2 - 5 years getting a graduate degree (e.g. a PhD). NOTE: once you are in graduate school in the sciences, you will almost certainly be paid to study. This will either come in the form of a teaching assistantship salary where you will supervise laboratory sessions for younger students in the early years of university, or in the form of a grant to work in a research lab with a senior professor.
A scientist can probably "sign on" to try finding a cure or cause of an illness as early as first or second year university. He or she could get a summer job, or a part-time job working in the lab of a senior scientist that is researching the cure for a disease. I know lots of young people who are doing this, and I did it when I was in university. Some very bright kids do this in their final years of high school.
Most scientists will start the process of finding a cure or cause of an illness or a disease by applying for a grant to obtain money to carry on the research. Such a scientist would probably be employed as a professor or researcher in a university, a hospital or a corporation involved in health research. The grant usually comes from the government, but it can also come from a private charitable foundation or society (e.g. the Canadian Cancer Society). Often the researcher will apply for numerous grants from many sources because this kind of research can be very expensive. In the grant application the scientist must explain exactly how he or she plans to conduct the research about the illness, and who else he or she will be working with, what kind of equipment they will use, etc. All this will get reviewed by other scientists and checked for feasibility before the scientist is awarded the money to do the research.
To get an idea of how much money is involved, some basic molecular biology studies get about CA$80,00/year in funding. Others need about $500,000/year because they require special equipment or have to pay for special scientific services. Yet compared to some other areas of science, such as high-energy physics, these would still be pretty cheap types of experiments. The researcher will have to prove himself or herself by showing results after a few years or the funding will be cut off. Once the research reaches the phase where a drug has been found, and human testing is required, the costs really get high. Drug trials cost tens of millions of dollars a year for staff, services, testing and all of the patients's expenses during the trial.