Choosing a "starter" telescope can be difficult, because there are so many choices and opinions.
You can google "buying a telescope" and get some basic advice from, e.g., Orion Telescopes or Sky and Telescope magazine. But what I'm going to give you here is advice that is even more basic. I'm going to try to get you to understand what your initial goals and ambitions are.
There is a saying that I have heard attribed to the Japanese: "The Three Whys". To understand someone, you have to ask "Why?" three times. (Once to start, then twice more in reply to the answers.) Before you ask "What sort of telescope should I buy?" you need to understand why you want a telescope.
Seriously. Why?
Alas those photos were made with time exposures and your eye does not "integrate" light like a camera. You will never see those colorful nebula clear and full of color (for dim objects, your eye only sees in black and white). Also, many (but not all) of the photos in astronomy magazines are of objects that appear very small: most galaxies are just tiny smudges to the eye (there are exceptions: the Andromeda Galaxy is larger than the full moon in binoculars.).
Related to Why do you want a telescope? is What sort of astronomer are you -- or think you will be?
One can divide amateur astronomers up in various ways, and their telescope choices differ. ("For a child" is a special case and I discuss it later.)
One division is between people who just want to observe and people who are interested in telescopes as devices. This is akin to the difference between car-owners who just want transport and car-people who read Road and Track and like to tinker, or between ordinary photographers and people who process and print their own images.
Another division is between people who like to look at and perhaps study in detail showpiece objects like the moon or Jupiter or the Orion Nebula and people who like to "test their observing limits" by trying to see difficult objects. This dichotomy is similar to a person who likes nature, but is happy with simple observation of showy things like redwoods and tigers, versus a "birder" intent on augmenting his life-list.
Another division is between people forced to observe in bright skies near cities, versus people fortunately rural. And related to that is the split between people who observe from near their dwelling and those who travel to better astro-locations for observing.
Think about these different types and decide who you are. Then keep that in mind when shopping for a telescope.
Important in your choice of telescope is where you plan on observing from. There are two aspects here: Is the site a good one for astronomy, i.e. clear dark and with steady air? Do you have to transport your telescopes there and set them up?
Although I now mostly observe from sites that I have to drive to, I recommend that a beginner observe from his home location. If that means, e.g., foregoing dim objects because your sky is bright, so be it. It is more important that you actually use the telescope than that your observing conditions are excellent. If you observe from your back-yard, or balcony, or rooftop, you can try observing most every clear night, without having to weigh "Gee should I drive to the observing site? The weather forecast is iffy."
Buying a telescope for a child is challenging. Your motives are excellent: Interest the child in science and engineering, let them explore some aspects of nature, offer a gateway to physics and mathematics. (And get them outdoors and away from television.)
But at the same time, you have to consider the child's knowledge, abilities, and patience.
I received a gift telescope when I was thirteen. Much younger than that and I think the telescope should be considered as a shared one, with the parent actually in charge.
Children are neither as patient or dextrous as adults. This makes fiddley mechanisms difficult for a child. And a telescope can be very fiddley.
This is expecially true for telescope mounts: a telescope magnifies the view a lot, and so it magnifies all of a telescope's shakes a lot. Further, the earth is rotating, so you have to keep nudging the telescope to keep it pointing at an object. If it is not easy to point the telescope and actually succeed in getting an object (like a planet) in the field of view, or if you cannoit keep it there because the mount mechanism is "sticky" or the telescpe is not balanced, a child will quickly become frustrated.
So my first rule for children is to acquire a very stable and easy to use mount. This implies a fairly heavy mount, which may also be a struggle for a child. But better heavy (important only twice per night) than shakey (frustrating all the time).
There are probably amateur astronomers near you. They may hold regular "star parties" where they bring telescopes and observe and mingle. If you can find such an event, it is an excellent way to learn about a) various types of telescopes b) the variety of objects to look at. Visiting a star-party and chatting-up the people and sharing views (just politely enquiring "what are you looking at" will usually get a look-see) is more valuable than any amount of reading articles like this one.
My local astronomy group, the San Jose Astronomical Association, has a "loaner" program that lets members borrow telescoeps from the club's inventory. If you have something similar in your area, this is an excellent way to "test drive" a telescope.
If you start out with a telescope, you will learn much more about your astronomical preferences than you could ever learn before you start.
So it is quite likely that your initial purchase will not be perfectly adapted to your eventual interests. (And even if it is, interests change over time.)
And maybe you will discover that astronomy is just not as interesting as you had thought/hoped.
For all of these reasons, it makes sense to buy an inexpensive telescope, planning on (if in fact you enjoy the hobby) upgrading later.
Building on your answers and thoughts to the above questions and discussion, I can make some suggestions about a first telescope:
You might decide that astronomy is not for you, and you'll save money. Even if you can afford an expensive telescope, using a small one for awhile will let you buy the correct larger scope later.
You do not want the child's experience to be frustrating -- kids have short attention spans even without obstacles.
But you do not need high powers, as are sometimes advertised. You do not need more than 80x or 100x to obtain fine views of the moon and the planets. And higher powers require sturdier mounts, almost require an electronic drive to "track" the stars, and are generally harder to use.
But this is less important for a small starter telescope. It becomes very important if you decide to buy a larger telescope, because above about six inches in diameter, refractors start to get very expensive and also the tubes get long and unwieldy.
Because of this, many people recommend a small refractor, say three inches in diameter, as a good beginner's telescope. This has much appeal. The telescope is simple to understand and use. The tube is effectively sealed, so dust gathers only wehere it is easy to remove. I personally like small reflectors because the tubes are shorter and I find the viewing position more comfortable -- but this is a matter of taste.
And remember to have fun! This is supposed to be pleasurable. If some aspect of amateur astronomy seems like work rather than fun to you -- don't go there! There are many different aspects of amateur astronomy, from planet-sketching to galaxy-finding. Some, like meteor watching, don't even require a telescope!
You can make a telescope by designing one and purchasing (or making) the parts and putting it together -- mechanical construction.
Or you can grind and polish your own mirror. Back in the 1950s, when I made a telescope, making your own mirror was popular. It was a major cost-savings back then. You could buy a 'kit' for making a six-inch mirror for around $12; a completed mirror that size was mebbie $60. (All in 1950s money.) Today, the economics are much less favorable. Still, there is the "pride of accomplishment".
Back in the first half of the twentieth century, there was a regular column on telescope making in Scientific American magazine. Edited by Albert Ingalls, these colomns were eventually collected into a trio of books. A bit later, Texereau wrote How to Make a Telescope. Both of these are fascinating reading -- Build a telescope mount using a Ford engine block? Cast your own aluminum parts in your backyard? -- but are mostly of historic interest.
I suggest that you do not try to make your initial telescope. Rather, buy an inexpensive one and use it. That will tell you:
and you can then decide, perhaps "I want to build my own telescope because I can't buy one that meets my needs".
If you read elsewhere at this website, you will learn that, along with making a telescope-mirror and telescope back in the 1950s, I have more recently constructed two other telescopes from store-bought parts, because I was not happy with the normal viewing posture. These two recent telescopes were the reslt of many years of experiencing and weighing various frustrations. I certainly would not have built a telescope that way in the 1960s, and I am sure that your frustrations will be different. So get some experience before you jump in with a "project" that will, among other things, take you away from observing for awhile.
Binoculars are in fact low-power telescopes. And I have seen "choosing a telescope" advice that suggests beginning with a pair of binoculars and learning the night sky.
I disagree. I think that was good advice decades ago, when everybody's sky was reasonably dark. But with increasng light pollution, more beginners are likely to be interested in brighter targets, like the moon and the planets. And binoculars are inherently low-power, while the moon and planets need decent magnification -- not huge magnifications, but at least 40x or so.
In recent years there has been a proliferation of "Go To" telescope mounts: align it, then tell it to "go to" an object in the sky, and motors in the mount point the telescope to the object.
This is definitely a convenience. It is helpful even for ojects that are easy to see and identify, like Saturn, because it can be surprisingly difficult to point a telescope at a dot in the sky using a "red dot finter" or small finder-scope. And in urban skies, it is more difficult to "star-hop" to point the telescope to a faint object, because if you cannot see faint stars, there are large areas of the sky that appear star-less.
Also, a "go to" telescope will track the stars as they appear to move due to the Earth's rotation. This is a convenience that means you don't have to keep nudging the telescope westward. It also means that you can leave the telescope for a minute or two and when you return the object will still be in the field.
On the other hand, such a mount will add greatly to your initial costs. And the mounts have their own frustrations, especially in the initial alignment steps.
For a first telescope, I would suggest spending your money on the basics and avoiding an electronic mount, because the additional cost is high. For a child familiar with computers, though, an electronic mount might avoid much mechanical frustration -- though it can introduce different frustrations..
Was this insufficient help? Did I miss a key aspect of telescope-buying that worries you? Feel free to contact me at astroayers@gmail.com