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.
Before you ask "What sort of telescope should I buy?" you need to understand why you want a telescope.
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.
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 regular 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 elk, 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 drive to, I recommend that a beginner observe from his home location. If that means foregoing faint 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, the weather forecast is iffy, and maybe the round-trip drive will be for nothing."
Perhaps you are fortunate and reside in a rural location where light-pollution is low and the Milky Way is prominent on clear nights. Good for you!
You can enjoy good views of fainter and lower-contrast objects.
You might lean toward a telescope that is good for low-power observing, perhaps with nebula filters. For such observing, your telescope's lowest power should be only about 4x or 5x per inch: e.g. 16-20x for a four-inch aperture telescope: this is much lower than is usually seen. You can find more discussion at my rich-field telescope page and my low power observing through a RFT page.
You can observe brighter objects like the moon and planets too. Such observing suggests higher powers, as high as 40x per inch. Getting both low powers and high powers with the same telescope requires some care in choosing eyepieces and in using "a barlow lens". You might consider starting with the low-power faint targets and, after some experience, adding higher-power observing of brighter objects. You might also end up with two telescopes, one more-suited for low-power RFT views and one more-suited for higher powers. But start with one.
Perhaps you live in the city or suburbs where light-pollution is a major problem and when you look at your night skies the Milky Way is invisible. Do not despair!
You can easily observe brighter objects. These include the moon and the brighter planets, as well as double-stars and compact star-clusters, like globulars. (Novice astronomers often underestimate the moon -- it can be a fascinating object for study. I still have the drawings that I made when I was a teenager.)
You should lean toward a telescope that offers somewhat higher powers (say 40x to 120x). With the higher magnification, you might consider the benefits of a clock-drive (see below).
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.
My parents gave me a telescope when I was twelve. (It was a four-inch Criterion "Dynascope" reflector.) 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 cannot keep it there because the mount mechanism is "sticky" or the telescope is not balanced, a child will quickly become frustrated.
So my first rule for choosing a telescope for a child 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 has a "loaner" program that lets members borrow telescopes 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 super-high powers, as are sometimes advertised. You do not need more than 80x or 120x to obtain fine views of the moon and the major 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 where 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 interesting reading -- 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 way back in the 1950s, I have more recently constructed two "folded refractor" telescopes from store-bought parts, because I was not happy with the viewing posture offered by normal store-bought telescope. These two recent telescopes were the result 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.
I discuss this more at my page on astronomical binoculars.
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, because it can be surprisingly difficult to point a telescope at a dot in the sky using a "red dot finder" 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.
On the other hand, such a mount will add considerably to your initial cost. 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 a "go to" 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.
Separate from the high-tech "go to" mount, is an ordinary mount with a motor (or motors) that will slowly move it to counteract the Earth's rotation. It doesn't help you locate an object, but you don't have to keep nudging the telescope to keep the object in the field of view. Such a mount is valuable if you are going to use relatively high powers, say on the moon or planets. This capability is often called "a clock drive".
Was this insufficient help? Did I miss a key aspect of telescope-buying that worries you? Feel free to contact me at firstname.lastname@example.org