150 and 400 years ago, two major events in the history of science occurred.
400 years ago, the telescope was invented and started to be used for astronomy. For $100-$150 you can now get a telescope far superior to what Galileo used to carry out a major revolution in our understanding of the universe. More in a moment.
150 years ago yesterday (November 27th), Charles Darwin's On the Origin of Species by Means of Natural Selection was published. Different major revolution in our understanding of the universe. You can read this for yourself. I don't actually recommend reading it unless you are really interested in history of science, and like Victorian-era writing. (If you like my style, you're a couple steps in that direction. My wife noted that I write something like Trollope, a prolific Victorian whom she likes.) We've learned an awful lot in the 150 years since then, and many things that were mysteries to Darwin, such as how inheritance occurs, are well-known to us now. Instead I'll suggest you read the evolution sections of modern biology texts. Two such texts recommended by my biologist friends are Futuyma's, and Campbell and Reece.
For the telescope side, since this is a time when a lot of people buy telescopes, a few words about how to shop for one. The first thing to do is ignore magnification levels. In looking through the atmosphere, you can only magnify so much before what you're seeing is the turbulence in the atmosphere instead of anything astronomical. A fairly good rule of thumb is 50x for each inch of the main lens/mirror, or 2x for each millimeter. Many telescopes advertise much higher than this, but the extra is somewhere between useless and actively harmful. Your children will not have much fun exploring atmospheric turbulence when they expected to be seeing the moons of Jupiter or the rings of Saturn (and even small telescopes can show these!)
The diameter of the main lens or mirror is more important 'score' than magnification -- this controls how much light the telescope catches. Bigger = catch more light. Galileo's telescopes were about 40 mm diameter (1.6 inches). The typical small telescope you can buy today is at least 40 mm, typically 60 mm lens. It isn't hard to find telescopes with a main mirror 150 or 200 mm (6 or 8 inches), though they'll be more expensive. (If you're especially gung-ho, it isn't hard to find 20 inch, 0.5 meter, telescopes!) Note that the size we're talking about is the mirror, not the length of the tube. Length of tube doesn't help you collect light. For lens telescopes (and eyepieces) you want to test that the images don't have rainbow halos. If they do, it's bad optics, not that the universe really does have rainbow halos around everything.
Last major element in telescope hunting is a stable mounting. If the image keeps shaking for some time (tens of seconds) after you make a minor adjustment, or a slight breeze passes, it's hard to pay attention to the astronomy when the universe seems to be swimming about before your eyes.
So, for not terribly much (this time of year you can often do better than the figures I quoted), you and your child can discover the universe and carry out a revolution in your own understanding. For myself, I prefer the discovery/revolution approach, over the 'hunt down other peoples' lists of objects'. For one thing, if you're in an urban area, many of those listed objects will be invisible to you.
On the other hand, many people really enjoy chasing the object lists, and astronomy clubs and societies often have guides and guidance on which ones to look for, and how to do it. Great accessory to the telescope itself is an astronomy club. A place to start your looking for local astronomy clubs is the Astronomy League, which also has a number of lists of things to go looking for. Internationally, you can try this, but their links seem often to be broken.
And Yet, We Continue Upward
3 hours ago