Non-US Citizens, including lawful permanent residents (Green Card holders), cannot legally vote in federal, state, and most local elections. Even registering to vote, as a non-US Citizen, is a crime and cause for imprisonment and deportation. It does not matter if someone tells you that you may vote. DO NOT VOTE until you are a US Citizen.
Be VERY careful at the Department of Motor Vehicles (DMV). The DMV and DMV forms routinely ask people to register to vote. Before you submit any form or application, make sure that you have not checked a box or accidentally signed to register to vote.
You may also receive fliers at home stating that it is your civic duty to vote. DO NOT VOTE until you are a US Citizen.
To be safe: DO NOT REGISTER TO VOTE. DO NOT ATTEMPT TO VOTE until you are a US Citizen.