When it comes to using organic means to control insect pests in the garden, you’ll need to think of your particular pests as being of one of two kinds: (1) sucking insects, and (2) chewing insects.
Sucking insects, like aphids, mealybugs, scale, and whiteflies have a little microscopic mouth part that allows them to poke into plant tissues and literally suck out the sugary creations that result from photosynthesis. There are many organic controls for sucking insects. These days I like to suggest Neem Oil-based product. Neem Oil is sold under many trade names. Ask at your local, independent garden center; they’ll have it.

Remember sucking insects, suck juices out of the plant. If you see holes in your leaves or pieces of leaves missing, it’s cannot be due to a sucking insect.
Chewing insects, however, as the name implies, actually chew on (or remove) parts of plants. Caterpillars and beetles eat holes in leaves; leaf miners eat tunnels in leaves; thrips eat the surfaces of plant tissues.
Here again you have a few options for controlling chewing insects organically. I suggest a Spinosad-based product for controlling chewing insects like: caterpillars, loopers, citrus leaf miners, cucumbers leaf miners, beetles, etc. Spinosad is sold under many trade names. I find the brand “Captain Jack’s DeadBug” effective and easy to find.

Before using any control, fertilizer, or anything that is supposed to “do something” to your plants, be sure to read the entire instruction label, AND do what the label says. Watch for the instructions about:
Safety concerns (even organic controls kill things)
When and when not to apply
How to apply
How much to apply
How often to reapply
Organic controls breakdown more quickly than synthetic controls (which is a good thing = they don’t stick around too long!), so they typically need to be reapplied more frequently, especially for heavy infestations. Read the label!
Also remember two other important points. First, just because you see holes in your leaves, doesn’t mean the critter that it did it is still present; they may be gone, and controls won’t make the holes in your leaves go away. Second, just because you see holes in your leaves, doesn’t mean an insect did it. Snails and slugs also damage plants, and using the above controls won’t deter them. Instead, I suggest and Iron Phosphate-based product like “Sluggo.” It’s organic and safe around cats, dogs, and neighbor kids. Read the label, and do what it says!