Science outreach encompasses many different career choices. If you are interested in such careers, you may choose to work directly with the public, either in an administrative position, or as a museum docent. You could also choose to work behind the scenes, by curating museum exhibits, designing interactive demonstrations, or creating connections between scientists and the public. You might also be interested in training scientists in science communication, to help them better participate in outreach projects. An added challenge is that many people refer to "outreach" in different ways. In some circles, it's known as public outreach. It has also been referred to as public engagement, service learning, informal education, and community learning, to name a few.
General Public Outreach: You can find public outreach coordinator jobs at colleges and universities (usually these jobs are in "centers" of outreach or teaching, but outreach responsibilities could also be found in a development office, diversity office, or public relations office), as well as institutions (national labs) and instrumentation groups (e.g. Chandra X-ray Telescope, for astronomers). These positions typically involve organizing large "fun" events or curating displays of artwork or educational demonstrations. The focus for public outreach is typically on creating a program that caters to what is popular in the field at that time. If you are interested in these jobs, you might also be called upon to run the social media platforms for a particular group.
K-12 Outreach: In contrast to public outreach, K-12 outreach involves much more collaboration with K-12 teachers. This can include working with teachers to create educational opportunities for the students (tours of campus/labs, presentation of educational demos, etc). Especially for older students (high schoolers), outreach should include professional development opportunities (research partnerships between high schoolers, high school teachers, and graduate students/faculty, panel discussions on pursuing careers in science, etc). K-12 outreach also typically includes a diversity awareness element, and outreach efforts can be focused on low-income schools and/or typically underrepresented students in STEM (women and minorities).
Phd required: recommended
Exhibit Curator: Depending on your field, a museum may require you to have a degree in museum curation in addition to a PhD. This is usually the case if your field requires the handling of delicate/rare/old/expensive exhibit materials. Museums also recommend that you get experience volunteering at a museum before applying for a job. It's fairly easy to volunteer during graduate school. Once hired, your job might entail 1) acquiring exhibit materials, 2) designing exhibits, and/or 3) evaluating the effectiveness of exhibits.