There are several options that will work for your situation. The first that comes to mind is creeping thyme, which loves full sun and is of course a non-toxic herb. It also flowers profusely. Lemon thyme has a variegated leaf. I am not sure how well thyme deters weed growth -- someone with experience can give you a better idea. Also, it is probably marginally hardy in Zone 4. Another non-toxic option for full sun is moss phlox (Phlox subulata) which is hardy to Zone 3 and evergreen. It forms a dense carpet and is therefore pretty good at suppressing weed growth.

For shade, the go-to is usually Pachysandra because it is evergreen, thrives in shade, and tolerates dog traffic well. It will spread slowly, but I don't consider it overly aggressive. It is toxic if consumed, but dogs aren't usually attracted to it. Ajuga is another (toxic) option. This will spread fairly quickly and might be considered aggressive if it isn't controlled by a boundary of some sort. It has nice stalks of purple flowers in Spring. Another (toxic) option is (Vinca minor) which is evergreen, has purple phlox-like flowers, and comes in a variegated cultivar called 'Illumination.'

You might wonder why I recommend toxic plants: Most plants have evolved with some degree of toxicity to deter consumption by mammals, birds, and insects. Toxicity is highly variable: a few leaves of monkshood can kill a child, whereas many plants contain mildly toxic compounds that won't do any measurable damage unless consumed in large quantities.

While avoiding the most deadly of plants (like monkshood) is certainly a wise strategy, I feel it is a bit naive to expect to be able to cultivate a landscape completely devoid of toxic plants. I assume you just want to avoid the plants with dangerous toxicity. If you properly supervise your kids, I wouldn't worry about the plants I suggested because they are not dangerously toxic. Your kids would have to consume a large amount of them for there to be an issue and I am sure you would put a stop to it if they were starting to eat it. Thankfully, most toxic plants do not taste good, so dogs and people tend to stay away from them.

Best of luck!

This post was edited by mistascott on Tue, Dec 18, 12 at 1:49

Oh, I completely forgot daylilies for full sun, the flowers of which are edible!