First off, one of the best things you can do to avoid using pesticides is to not plant too much of the same thing close together. If you have a lot of the same thing planted, it draws the insects to them - sort of like a crowd of people will draw more mosquitos than one lone person (unless you're me, then all bets are off!). If instead of long rows, bunched together of the same fruit or vegie, you'll interplant them, it helps to be able to avoid bugs.
In my Square Foot garden, I would plant a 4 x 4 square of corn - that does need to have plants next to each other, but then, I would plant something in the next 4 x 4 square that wasn't related to the corn family. Perhaps potatoes (which for the record, you can just dump your cut up "eyes" on top of some hay, straw or pine needles, lightly cover with some more of what you're using. As the plant grows, keep adding more of your cover material. When it's time to "dig", you pull up the plant and up comes the potatoes, nice and clean.)
I would only plant 2 or 3 tomato, green peppers, etc in each 4 x 4 square. Since I had 9 squares, I still could have a lot of the same plant, but they were spread out over several large squares.
My fruit trees, I would plant in small groups. Some trees need others to pollinate and bear fruit. But 2 or 3 apple trees isn't an acre of apple trees. The acre will draw more bugs.
The second thing to do is look at your plants each day, when you water them is a good time to look them over, pull any weeds and hand-pick off any critters. I would take a cup that I would cover with my hand as I went around, when I was getting "escapies", I'd go dump them in the chicken pen. (You don't water because of a watering ban at your place - just take the "u joint" off from under your sink and place a bucket under it, or run your rinse water into a dish pan to rinse your dishes; then use that water on your plants. I wouldn't use water that I washed my hands in after using the "facilities". You don't want any possible E-coli or other nasty bugs in your garden. Do not use water from underclothes, diapers or cloth sanitary supplies, but if you're just rinsing food before cooking, rinsing dishes, regular laundry then that water is safe to reuse. Also, I had a fish tank and when I'd change part of the water in it, I'd dump that on the garden too.)
The next thing is to make sure you don't leave any decaying fruits or vegies laying around. The decaying will draw bugs that feed on that particular plant. So pick up windfalls from under your fruit trees, and make sure you don't leave any vegies on the vine/plants that are decaying. Also, mulch any diseased leaves, don't just till it back into the soil near the garden.
Even having done all of this, you will still have some bugs. They are a fact of life. Sometimes you have an infestation in the soil itself. Then we have to get tougher.
There are several products that will cause the bugs problems, but are safe for people and pets, not to mention the earth itself. Here are three websites that will tell you more about different pesticides that work well and are safe :
I've put the links up to keep from redoing their excellent work.
One other thing is that there are things you can do to draw "good" bugs to your garden. Or you can purchase these and introduce them into your garden. Check out the seed catalogs or Google beneficial insects and you should be able to find places to buy such insects as Lady Bugs, Praying Mantis and other helpers.