Your two most common (and safest) natural insecticides are pyrethrum, which attacks the nervous system and works on contact and neem oil which works as an anti-feeding, and over time will affect the reproduction cycle of insects that choose to feed on your plants. Pyrethrums are common in insecticidal soaps and it's my go to product when I actively see pests and want something that's fast acting. I've been slowly getting away from neem oil because it can burn some finicky strains.
Food grade diatomaceous earth is also popular and unlike chemicals (be they organic or synthetic), it won't be absorbs by the plant, it doesn't impart an unpleasant taste or odor, and works by mechanical means so it won't ever cause injury or affect nutrient uptake on plants. Plus its used in biscuits and it's totally safe to eat. DE works by cutting the insects exoskeleton causing them to bleed and by getting into their pores and dehydrating them (think salt on a wound).
For slugs and snails, wrapping coper tape around the base of your stem and the contains is also an effective deterrent.
If your open to using synthetic pesticides, but want something safe enough for vegetable crops, but that will have no problem killing larger pests like grass hoppers, then I would suggest you look into a product called Sevin 5% Dust, chemically known as Carbaryl. You can buy a shaker can at Home Depot for around $6 or $7 dollars. You simply dust your plants early in the morning when they is a light dew on the leaves (otherwise mist them first so the dust sticks). Its very strong, so you only need to do a light dusting, and it's safer then many organics insecticides (like nicotine). However, you can only apply it 4-6 times and you should stop using it 3-4 weeks before harvest. It's been approved for roots, veggies, and even leaf crops like lettuce and lots of people on GardenWeb are raving about it. I just tried it for the first time this year and I'm slowly becoming a convert.
