Simple answer: you just pay Lakewood tax, since you both live and work there.
BUT, it sounds like you may work in Cleveland often enough that you may be subject to Cleveland tax as well. Generally, if you work more than 20 days*, in a year, in a city with a tax, you are required to file a return for that city. Your residence city will give you a credit, or partial credit for the tax you pay the work city**.
You may want to contact the Central Collection Agency (CCA), which administers Cleveland Income tax, for instructions and verification on filing. Phone 216.664.2070
*20 days is the general rule in Ohio, but each city may have it's own threshold.
**The city of Lakewood has a tax rate of one and one-half percent (1.5%), but only allows a credit of up to one-half of one percent (0.5%) for a tax paid to other localities. Since the Cleveland tax is 2.5%, you will effectively be double taxed for the days you worked in Cleveland.
