It is quite common for students to transfer from one school to another at the undergraduate level. In fact, some schools (many 2-year colleges) are designed to be transfer schools and offer only the first two years of undergraduate study. Their graduates often go on to do the last two years of a bachelor’s degree at another school.
The situation regarding graduate programs is more complicated due to the nature of graduate education (see Question #3, above). It is sometimes possible to transfer to another university after beginning a graduate program in the U.S. The policies, however, will vary by individual school; therefore, it is important that you inquire of the university where you plan to transfer as to exactly what courses they will accept for transfer. You should make sure to inquire about the general transfer policy as well as the specific policy that pertains to the school from which you will be transferring. It is likely that some of your previous graduate credit will not be accepted. You should choose the school you are transferring to carefully, and you should probably discuss this with your current graduate advisor.