First, make sure you’re actually having a “problem”. It’s normal for a pest to wander in occasionally. If you’re seeing multiple cockroaches in your home each day, you have a problem. If you see one in your home each month, you probably just have a traveler stopping by.
Secondly, ask yourself if you have small, medium, or large cockroaches. Small would be slightly longer than ½ inch or less (as an adult). Medium would be approximately 1 inch long or slightly greater (as an adult). Large would be approximately 1 and1/2 inches long (as an adult). Most adult cockroaches have wings, so make sure the insect you are judging has wings to know what size you have.
After that it depends on what region of the country you live in to determine what species of cockroaches you have in your area. This can probably be found out through your county agent, your state’s department of agriculture, or the closest agricultural university or college.
