Many state and local governments are organized similarly to the federal government structures. Before you apply, though, consider reviewing ESAL's Civics 101 highlighting basics of local governments. Research some of the current work your local governments are implementing and follow work that interests you.
If you do not see an opportunity that directly aligns with your interests but want to stay informed about openings with governments, consider researching your state or municipality's civil service exam. By taking the exam, what opportunities open up for you? People who take and pass these exams and placed onto civil service lists, which make you eligible for various openings in local governments.
Review and update your CV or resume to align with what your government is seeking. Once you identify an opportunity, review the tasks listed in the job description, identify your relevant skills and experiences, and feature them in a concise resume.
Related, customize your cover letter to explain how your STEM training uniquely prepares you for filling a specific role with your local government.
When you can, seek another opinion. Ask a colleague to review the job description and your application materials together.