Syringes are considered Sharps. Sharps are medical devices that can cut or puncture the skin. Proper disposal rules and regulations can vary from state to state and from locality to locality. The website https://safeneedledisposal.org/ provides some suggestions that may be helpful to you.
When visiting this website you will note that there is an informative one minute video, as well as a zip code finder, located at the top of the screen. By using the zip code finder you can find resources for your particular location.
Another excellent source of information on proper disposal of Sharps and home medical waste is your local trash hauler.
For more information on disposal of unwanted, outdated and unused medications, contact your local police department. Many police departments have receptacles in their stations and hold collection events.