A lot of what we do is spelled out on the AHA website in the Government Affairs section: http://www.homebrewersassociation.org/pages/government-affairs/statutes
In terms of homebrew legalization, the process works like this:
We are contacted by local homebrewers who are interested in changing the laws in their state.
We can provide history of past attempts to legalize homebrewing based on whatever records we have on file.
We can help line-up allies within the brewing community.
We also advise homebrewers through the legislative process, provide samples of statute language from other states, who to involve in the process, how to avoid creating enemies that can sink the process, etc.
When needed, we rally the troops to contact politicians to garner support for bills, including all AHA members within the state, Brewers Association brewery members, and Support Your Local Brewery Members as well as posts to the AHA Forum and TechTalk. We include other influential people within the brewing community, such as The Brewing Network, Basic Brewing Radio, and Beer Advocate in our Action Alerts. We also use social media, including twitter (@AHADirector) and the AHA Facebook page to spread the word.
For both Oklahoma and Utah, we sent press releases when the legalization bills were signed into law, though our media database is primarily beer and food writers.