Most people believe there is a long list of insects that invade homes and can potentially damage our valuables or even hurt us. The truth is the list is very short and in modern society these species pose little real threat to us or our homes. Oftentimes the drastic steps we take to avoid these species are potentially more damaging than the insects themselves. In other words "the cure is worse than the disease".
I've been answering questions about insects from the general public for over 20 years (see my bio here) and the following eight are the ones most often found in homes. These eight represent the vast majority of insects that ever enter homes and none pose a significant threat if reasonable levels of sanitation are maintained.
The Big Eight Household Bugs
fleas (only if pet dogs or cats are housed indoors)
Obviously individual pests on this list will vary in importance depending on where you live. For example, cockroaches are more common in warm climates whereas silverfish need dampness to thrive but small nuisance ants can occur almost anywhere. Take a look at the individual pest pages by clicking on the links above. Each set of pages describes the pest's life history and some options for control when needed.
What about insecticides?
I almost never suggest that people use insecticides indoors to control household pests because with conventional pesticides any potential benefit is far outweighed by potential hazard. However, there are now new natural insecticides made from plant oils (so called botanical pesticides) that are safe enough, and effective enough, to be recommended in situations of even severe infestation. The main downsides of botanicals are somewhat higher cost and limited availability. See the individual pest pages cited above. A good selection of plant-based botanical insecticides are available here.