By attending these institutions you are ensured that the curriculum for Space programs offered will conform with guidelines NASA finds acceptable. Many schools offer degrees in technical fields, math, and science. You will need the grades to graduate into a good Master of Science program. Obviously you will need to center your curriculum around science and the technologies. There are many degree options.
Space is a multinational and multicultural-cultural operation.
Students who did not explore career possibilities until their senior year could miss this opportunity. There isn't one particular type of work experience that NASA is looking for. Everyone should have summer jobs. You need some sort of experience. This will also help you to understand what the particular companies are looking for in terms of hiring policies and experience levels.