The answer jbatista provided is good, if on the conservative side. "Turn Left At Orion" is an excellent recommendation. Install Stellarium on a laptop or iPhone. Get the Pocket Sky Atlas for a dead-tree format of the map.
Here's a different view, aiming to shoot a bit higher:
If you do get binocs, do not forget that you never do astronomy hand-held. The binocs must be sitting on a tripod.
But for the price of binoculars, you could get the Galileoscope instead:
The objective lens is a surprisingly good achromat doublet. The instrument accepts standard 1.25" eyepieces. Plus, the 6 year old and the dad have the opportunity to build the scope together and, therefore, learn how it's made. You will need a photographic tripod to keep the instrument in a fixed position while observing, see if you can find a non-wobbly one for cheap at a garage sale (it's important that it doesn't wobble).
I'm not a big fan of the numerous "starter scopes" or "first scopes". While cheap, they are often poor quality. The beginner has no chance of choosing correctly between the good and the bad ones. That leaves a big hole in the \$100 - \$300 interval. Below that, there's binocs and the Galileoscope. Inside that interval, there's a whole lot of hot air and deceptive marketing.
Here's a possible exception - the Orion SpaceProbe 3 Altazimuth:
I say "possible" because I haven't looked through it. But it seems promising because it has a long focal ratio: it's an f/9 scope. It's hard to screw up such a long focal ratio. You could plop in a cheap eyepiece and chances are it will work okay.
The other "beginner scopes" are very short, very fast focal ratios, f/4 or even shorter. It's hard to make such fast optics, or if you do make them well they're not cheap. My feeling is that those scopes are rushed out jobs, because of the low price. But even if they are made at a high spec, at f/4 focal ratio few eyepieces will work well; all the cheap eyepieces will be under huge pressure to perform, and will fail, no doubt - the image will be full of aberrations. Only very high-end eyepieces (that cost several times more than the whole scope) will perform well and will not add to the already existing problems of the cheap fast mirror.
So, if you do get a "beginner" scope, look for the ones with a long focal ratio, like f/8 or maybe even higher. They are far more forgiving on the eyepiece, and are far easier to make to a higher standard. They are not guaranteed to be better, but chances are higher in that region.
Finally, if you feel this will be a long time hobby, there is a scope that could possibly last you a lifetime (unless you're bitten by the aperture fever bug) - the 8" dobsonian:
It's above your stated price limit, but you said it's okay if it's for a good reason. Well, this is a scope that will show you a lot, for many years to come. The 6 year old will need a stool to stand on, when looking at objects overhead, but otherwise there should be no problem. It's much bigger than the beginner scopes, so think about it, and decide whether this is okay at this point or not.
Some more advice:
Magnification means nothing. Don't get fixated on huge magnifications. Even if your car can do 300 km/h, you don't drive all the time pedal-to-the-metal. You go slowly to the grocery store. You go faster on the highway. You go very slowly when parking the car. Different speeds for different needs. Same with scopes and magnification.
The atmosphere will limit your instrument anyway. At 150x magnification, not much of a limit. At 250x things get sketchy. At 400x you need really good skies, or else the image will look like a pile of wobbling jello. This is called "seeing", and it affects all instruments.
You change magnification by changing the eyepiece. If F = focal length of the primary mirror or lens, and f = focal length of the eyepiece, then:
Magnification = F / f
Ideally, you'd need several eyepieces to cover a wide range of mags. This is much less of an issue with small scopes; two eyepieces will suffice. With larger instruments, eventually you may feel the need to build up a collection.
What does matter is aperture (the diameter of the primary lens or mirror). That is the main limiting factor of a scope. A bigger aperture will always show you more. But at the same time it's more expensive and harder to wield.
With a small aperture, high magnifications are pointless - the image looks like a 50 x 50 pixel photo that was blown up a lot in Photoshop. With a beginner scope, it's usually pointless to go above 100x or so. Larger apertures allow higher magnification, but then seeing may or may not limit you, depending on current weather conditions. A huge dobsonian may allow 1000x mags, but then you need once-in-a-decade seeing conditions in a very good location; most of the time, under normal seeing conditions, it will chug along at 1/3 of that.
As a general, not very strict, rule, the maximum useful magnification is twice the aperture measured in mm. The Galileoscope has 50 mm (2 inches) of aperture, so the max useful magnification is around 100x. A 4 inch (100 mm) scope goes up to 200x. The 8 inch dob goes to 400x. But all this depends on seeing and a whole lot of other factors.
You will spend a lot of time at low magnifications, too. E.g., when observing large objects that you want to grasp at once. So don't forget about the low end, it's important.
Aperture also affects the brightness of the image. Small aperture will not be able to show you the very faint objects. Large aperture captures more light and you see distant galaxies and quasars and whatnot.
Light pollution might be an issue. It's not a problem for bright objects like planets and double stars, but it is a problem for nebulae and galaxies. Dark country-side skies are much better than the washed-out urban ones.
Clear skies to you!