Aphids, spider mites and scale insects are the most common insect pests on houseplants. All three damage plants when they feed on plant sap. Aphids may also infect plants with plant pathogens or inject plant toxins that cause growth deformities.
The most important step in control of these pests is early detection. Check plants often and treat as soon as pests are found. Scout for aphids along stems, especially at the tips, flower buds, and leaves. Cast "skins" and sticky honeydew are early indicators of an aphid infestation. Look for spider mites also on the undersides of leaves. Use your hand lens to carefully search a few leaves on each plant.
As time goes on you'll learn which of your houseplants are more susceptible so you can concentrate on these in the future. Look for scale insects along the stems and leaves. Scales are generally larger than spider mites (except the crawler stage) and should be easier to find.