In addition to the AAS John Bahcall Public Policy Fellowship, there are several other options for AAS members to enter the world of science policy. A subset of available internships and fellowships in science and technology policy are listed below, but sciencepolicy.us maintains a great, longer list as well.
A 12-week program for graduate students or those recently receiving their Ph.D. This is a popular entry-point into science policy; there are multiple sessions and deadlines each year.
AAAS fellows are placed in various executive branch agencies such as the State Department, EPA, NSF, etc. The AAAS also sponsors directly two Congressional fellows who work at Congressional offices or committees. In general, you must have defended your Ph.D. by the application deadline. However, some AAAS fellowships like the Mass Media Fellowship, are for undergraduate and graduate students. Deadline in usually Dec 15, for fellowships that start in September.
The American Institute of Physics sponsors one State Department Fellow, and two Congressional fellows, both under the umbrella of the AAAS S&T Fellowship program, but all selected and sponsored by the AIP. The AAS contributes to the funding of the State Department Fellow. AAS is a member society of the AIP, and AAS members can apply for both fellowships. U.S. Citizenship and a Ph.D. is required. Deadline is Jan 15, for fellowships that start in September.
Members of the American Geophysical Union (which has many members in planetary and space science), can apply for the AGU's congressional science fellowship, which is part of the AAAS fellowship program. The program is aimed at early-to-mid level scientists with doctorates, but there are no specific requirements on educational or career level. Deadline is Feb 1 each year.
Members of the American Physical Society may apply for the APS fellowship, and it is possible to apply for the AIP and APS congressional fellowships with a single application. Ph.D. is in general required, but can be waived in certain circumstances. Deadline is the same as the AIP, Jan 15.
For undergraduates who have completed their junior year, summer and autumn internships exist, and they last for 10 weeks. Deadlines varies with start date.