You don't have to worry about city taxes in most parts of the country. About the only universal type of tax imposed in cities is sales tax, and that's usually collected by the state, even if part of the money comes back to your city. Fewer than 200 cities of any size tax income, as of 2012. A few impose taxes on people who work in the city but don't live there. Taxes on income go by various names, such as wage taxes, payroll taxes or local service taxes, and they're collected in different ways.
Call your city hall and ask for a city clerk, city treasurer or similar official who would know about city taxes. It may take some searching, because not all cities have the same titles for financial officers, but every community has a top financial officer. Ask your boss, especially if you're new to the area, because most local taxes are collected through an employer.
Visit your city's website and use the search bar to find terms such as "city taxes." Doing that on the Columbus, Ohio, city website for example, will take you to the Income Tax Division. In San Francisco, you'll be directed to the city treasurer. Use the website to get specific tax information or the name of somebody you can call for information.
Refer to the Tax Foundation, a nonprofit organization that compiles information on all sorts of state, county and local taxes. Look up your state and city on its website. You'll generally find complete information on any local taxes and how they're collected. Use this if you're thinking about moving and want to check a new area.