You need to match the pesticide to the pest. My all time favorite is BT. It gives the bugs a belly ache so bad they just stop eating and starve to death. it's a biological so it needs to be stored kind of cool and dry or it looses it's effectiveness. But there are different BT's designed for certain critters. You have to get the BT good for the critters you are after. A very popular one targets certain caterpillars. I use that one mainly for the caterpillars that attack cabbage, broccoli, and other certain plants. I use another one that targets mosquito and some fly larva. Those are called "dunks" and work great in rain barrels and such to kill the mosquito larva, but it does absolutely nothing to cabbage moth caterpillars. This means the BT for caterpillars does not harm critters I want for pollinators. And it does not harm people.
Pyrethrum is a natural pesticide made from chrysanthemums and is considered organic. That doesn't mean it won't harm humans if used incorrectly. This one will kill pollinators. A poison is a poison, whether natural or manmade. If you use that one read the label carefully. There is a group of synthetic pesticides based on this called Pyrethroids but they are not considered organic as they are manmade. These often cause confusion. Actually you should always read the labels carefully whatever you use.
I'll let others discuss some of the others and hopefully tell you about some of the tricks we use, like maybe using egg shells against slugs. For a lot of these things you don't get 100% results, they reduce the damage to more acceptable levels.
Okay, you don't want to keep them down, Kristie. You want those vegetables to grow and thrive!
I know. You are talking about bugs. You want your plant allies to be safe from the pests. Nothing quite as disappointing as to go to great bother to have a vegetable garden and see it "harvested" by pests. They will take full advantage all too often.
About the simplest effective way to get rid of something like aphids is to spay those soft-bodied creatures with soap. Dish soap cleans things that come in contact with our food. It isn't necessarily harmless to living plant tissue. At 2 or 3 tablespoons of a simple dish soap, per gallon of water, it can kill aphids but, out of additional kindness to your plants, should be rinsed off them before they have problems with hot sunlight. The same rinsing is probably a little less important for the soap made for use on plants.
The Bt sprays were of great help to me to combat things like cabbage worms and potato beetles. The one "designed" for beetles turned out to be a GMO product. Who knew!? Anyway, the beetles took advantage of that spray being taken off the market. Spraying them with pyrethrum killed them but it had almost zero residual effect. Showing up every few days to spray eggplant, for example - I had dead bugs all over the ground and living ones, killing the plants! Then I tried spinosad.
Spinosad kills not only beetles and their larva but cabbage moth caterpillars. So, the Bt spray left on the market for them, gets little use in my garden these days.
The pyrethrum is still useful for things like stink bugs. Critters that hide from me and the spray and I worry are just going to migrate from one squash plant that they have been chewing on to another squash plant.
Insecticidal soap, spinosad and pyrethrum - the big 3 for my vegetable garden
Steve
Edit: oh and slugs: first of all, I don't mind seeing some bug spray on the ground around the plants. Spraying just about anything I've used seems to deter slugs. Often they are migrating from shelter of some sort, like grass or bushes. They go back and forth, overnight. To really kill them, slug bait (iron phosphate). They eat it but it melts easily in the rain or irrigation water and it's expensive.