Figuring out your local polling location can be accomplished through your state government's official website. Some websites have links for local voter-information pages, such as CanIVote.org or Vote411.org.
Can I Vote is run by the National Association of Secretaries of State and provides a link to all 50 states' government websites. Click Find Your Polling Place and pick your state from the drop-down box, and you will be taken to your state's voter-information page.
The information you must provide to find your polling place varies by state but usually includes some combination of name, address, county and date of birth. The results typically show your polling location, along with your party affiliation and whether you can vote in your party's primaries.