I like to fill a spray bottle with cold water and a few drops of dawn dish soap. Then I spray the undersides of the leaves early in the morning. For most of the year this is all I need and my backyard looks like an untamed field. Try not to get worked up over a few holes in the leaves. If it's just a bug here and there, and you don't do anything rash, predator bugs will soon come and take care of the leaf eaters.
Now if you start to notice an infestations or your worried about excessive damage, lightly dust the plants in food grade diatomaceous earth. This works best early in the season and no matter how sensitive your plant it, it will never have a bad reaction. The downside to diatomaceous earth is that it's non-specific, meaning it will kill any small pests, even friendlies like spiders, lady bugs, and bees. To address this I like to follow up with a pyrethrum based insecticide. This stuff is very safe around pets and people, it breaks down fast, and kills on contact. Also while it's active, it's a big pest repellent. If any pests are still dumb enough to nibble on the leaves, they get a double slap in the face. The DE works by acting like glass, cutting them up and dehydrating them, while the pyrethrum attacks the nervous system.
Neem oil is okay, but it's not my first go to product. I know lots of growers use it and love it, so it's worth looking into. The way it works is by effecting the reproduction cycle. That means pests have to consume it for a longer period of time before it really starts to work. Its not really a contact killer, but it does kill some pests because the oil (and this is true with any oil based product) can suffocate small pests. To make the most out of neem, apply dish soap to break up the oil, which also helps it soak into the plant, and apply it once a week for several weeks.
Another option besides neem is Sevin 5% dust. It kills small insects as well as larger ones like grass hoppers and beetles. Its safe for veggies and fruits, as long as you follow the directions on the bottle. Its also safe around people, dogs, and cats. When nothing else seems to work, I use this. To be safe, I only apply it no more then two times a year and only during the veg. Its not organic. It's a true pesticide (not an earth friendly insecticide), but it works so good that you don't need to keep reapplying it.