Luckily, if you're renting a place, you'll probably have some reliable sources to contact about pest activity. For one, you might be able to contact the former tenant. If he or she is still living there when you tour the place, take that opportunity to ask about any pest problems. Also, notice how clean the person is. If it looks like the garbage piles up often, bugs could be frequent visitors. You don't want to inherit a bug problem due to another's lifestyle.
In addition to the former tenant, you can also try asking the neighbors in an apartment building. If you take a few minutes to knock on the neighbors' doors and introduce yourself, you can ask if they've ever had pest problems, which will give you a great indicator about whether you'll have pest problems.
Talking to former tenants and neighbors will give you an opportunity to ask if they have any other complaints about the rental property. Of course, this is also just a great excuse to make friends with your future neighbors.