Question

what are methods that local governments use to pay for the services they provide to their citizens.

Answer 1

Taxes

Answer 2

Real estate taxes, in states where those are kept by the county. User fees, for example business license fees or permit fees for construction. Revenue from fines and confiscations resulting from law enforcement, for example the revenue from traffic and parking tickets issued by local law enforcement, or the result of a confiscation, such as a car bought with money earned from selling drugs, which is confiscated as part of a criminal conviction (e.g. because the car was used in the crime). Some local governments even impose taxes on income, when that is permitted by the state constitution.