Whatever you do, DO NOT get your child a department store telescope with 60mm aperture advertising 600X magnification. It will be an exercise in dissapointment and frustration, and will more likely than not kill their budding interest. Cheap optics, and a very shaky mount that means you can't view anything at high power. Usually binocs would give a better view.
Anyway, for a casual observer who wants to view the moon, hunt for comets, see beautiful star fields, and catch a few of the more obvious nebulae, binoculars are the way to go. You want low power (<=10X magnification), and large objectives (>=50mm). Mine are Celestron Ultima 8x56, and i love them. Celestron makes top-quality binocs, using porro prisms (rather than the more compact roof prisms seen in more compact binocs, not as good for low-light environments) made of BAK-4 glass (high image quality). The optics are all fully multi-coated (antireflective coatings on front AND back of optics, minimising reflections). Here is my Celestron astronomical binocular pick (and they can be used for other purposes, too...):
Celestron OptiView LPR 10x50 mm Binoculars $80 at Amazon.com
10x power is a bit too much for some people... your hands' shaking may make it hard to see smaller objects clearly. "LPR" stands for light pollution reduction; they have anti-light-pollution filters built right in! Actually, these binocs use the cheaper Bk-7 glass prisms, but $80 is a very good price.
If you have a richer budget, I'd recommend these, although they won't be available til early 2006:
Celestron Ultima DX 8x56 Waterproof Binoculars $290 at Amazon.com
You won't see any details of planets through binocs, and you'll see only a little detail in even the Orion Nebula. If you want to see the rings of Saturn, the polar ice caps of Mars, the clouds of Jupiter, or resolve much detail in nebulae, you need a telescope.
Get a reflector (mirror is the objective), rather than a refractor; you can get more aperture (light-gathering ability) for less money. The primary function of a telescope is to gather light, not magnify. I'd also recommend something on a Dobsonian mount, a simple rocker box, essentially. It is inexpensive and very stable.
Orion makes some very nice beginner scopes, and they are inexpensive.
Here is the cheaper:
SkyQuest XT4.5 Dobsonian Reflector $199 from Orion Telescopes
If your budget can afford it, I'd REALLY recommend instead:
SkyQuest XT6 Classic Dobsonian Reflector $259 from Orion Telescopes
The 6 inch mirror really gives you a lot more viewing/resolving power than the 4.5 inch mirror.
Both come with two plossl eyepieces (one medium, one high power), finder scope to assist in aiming, and "Starry Night" astronomical software, which I enthusiastically recommend as among the best educational software available.
From this page, you can compare the two scopes side-by-side.