Catch that bug!
Inexpensive bug catchers make observation easier. Simple plastic boxes with removable lids can be temporary hotels for crawly critters. Some bug catchers have magnifying lids that make detailed observations easier and protect little invertebrates from overzealous fingers. Common city bugs you can find are potato bugs, shiny black beetles, grubs, centipedes, flies, caterpillars, butterflies, moths and spiders.
My kids love "The Bug Book & Bug Bottle" by Hugh Danks, a bug ID book accompanied by a bug bottle, a simple magnifying glass and a kid-sized journal to record what you find. You can also order a wide variety of inexpensive tools at Acorn Naturalists and Insect Lore.
Another good field guide is "Insects of the Pacific Northwest" by Peter and Judy Haggard. We also use a spectacular government website on local caterpillars, butterflies and moths to identify what we find, learn what they eat, and see what habitat they live in.
One caveat, though: Teach your child to stay away from bees and wasps, and do not hold hairy caterpillars in your bare hands until you know if the hairs can irritate the skin (we found this out the hard way!).