In the past I've also used diluted Dr. Bronner's but this year I'm trying out clove oil.
clove oil will kill the plants. it's one of the main ingredients in our non-selective weed elimination products.
i echo the other sentiments about monitoring and eliminating. when watering in the morning i give the leaves good blasts with water to hose off eggs/aphids/etc. aphids can get outta control mighty quick because they ARE BORN PREGNANT. i shiitake you not. they've found secondary embryos inside the existing ones!
beneficial insects are very popular at my workplace, but are expensive. they will control, but not eradicate the problem. since most-(not all) but most pests have a soil stage, i suggest beneficial nematodes:http://www.wormsway.com/detail.aspx?t=prod&sku=BN700
the great thing about the nematodes is that SO MANY insects have that soil stage. i've used these for flea and tick control.
i also suggest you buy them in a 6 pack to save $, because you need to repeat applications every 8 weeks. you can store them in the fridge too.
keep your gardening space dark at night so moths won't lay their eggs on your leaves. if you cannot cut out the lights, get a row cover to keep moths away.
like kittee mentioned, i'm not a fan of neem oil because it can clog the stomata of the plant. a cheaper solution is hot pepper wax. i also know the community garden makes a slurry of dead insects-(not vegan!) and sprays the plants with them. problem insects won't hang out where they smell dead bugs. i've never tried this.
make your garden inviting to predators of problem insects: toads, frogs, and birds oh my! hummingbirds love insects. wasps are beneficial in the garden, and frogs/toads will happily nom those bugs and slugs.
i also like safer brand products. they do have chemicals, just not as harsh. and i've learned to love sluggo-(iron oxide slug bait) and BT. BT is a bacteria that kills caterpillars. i've lost too many crops of kale and tomatoes in the past to the looper and hornworm. it really works.