I got involved with science policy primarily through the graduate student council (GSC) and MIT. I helped co-chair the federal affairs subcommittee and planned/attended two advocacy trips to Washington DC.
The American Association for the Advancement of Science (AAAS) runs many good programs around science policy which can mostly be found through their government relations page and their advocacy toolkit. They also run the Science & Technology Policy Fellowships that place PhDs in congressional and federal offices for 1-2 years.
Science Insider (also from AAAS) is their science policy blog on the Science Magazine website.
Union of Concerned Scientists is a nonprofit group that analyzes problems, proposes solutions and covers policies.
The science section of Vox isn’t specifically focused on science policy, but the website’s strength’s at breaking down complex policy shines through in their science coverage.
If you want to follow federal legislation, you can sign up at Govtrack and have info emailed to you on the legislators and topics that are relevant to you. The summary page for each bill will tell you the text, co-sponsors, the stage its at, and a rough estimate of its odds of passing.