I recently took the plunge and purchased a compound microscope for home/homeschool/fun use. I also have kids of this age (8, 6 and 5 now) and wanted them to have a nice scope for the future. The scope itself was only $200 but the additional things I wanted to go with it add on to the price, so be aware of that. I'm a scientist by profession, but not the kind that looks through microscopes professionally so I was able to not agonize over decisions (I can't pick a telescope, for comparison). My comments are limited to compound microscopes, as I haven't looked into stereo microscopes (yet!). I will second the idea to not "chase" magnification. You will be disappointed in the results past about 1000x unless you want to bump up an order of magnitude in cost.
A microscope is like a SLR/mirrorless (i.e. swappable lens) camera; you can think of the body and lenses as somewhat independent. In fact, if you look at a site like amscope, they have a handful of bodies in a myriad of configurations, so it's best to step back and think of them separately.
Lets start with the microscope body. It provides a number of features necessary for looking at things. First, the stage is where the sample sits to look at; it moves up and down (usually) to focus, and could also have the ability to move laterally to change the field of view (rather than just manually moving the slide). Second, it has some ability to illuminate the specimen. That could be a LED/bulb below, with or without an iris, or something as simple as a mirror to direct ambient light up through the sample and into lenses. Sometimes they also have a light above the sample to illuminate something opaque. Third, it has one or more places to look through (eyepiece holes). This could just be a single one, and it could be straight up or at an angle. There could be two, to be used separately (one angled and one vertical, sometimes called a "teacher's eyepiece") or together (binocular eyepieces, which usually have diopter adjustment). There could also be three (this is what I have; binocular plus a vertical third eyepiece). Finally, they have something to hold the objective lenses and quickly swap them out (the nose, which can rotate in the different lenses). There could be three-five spots there, depending but 3 or 4 is most common especially at this price point.
I would highly recommend a moveable stage. It really makes looking through the scope so much nicer, especially under higher magnifications. It is the difference between a "nice" and a "cheap" feeling scope. Second, make sure that you have coarse/fine focus, and I recommend seeking out a coaxial focuser. This is where there is an inner wheel and an outer wheel which are geared to turn at different rates, but together. It's a pain to have separate coarse and fine focusers and to run out of fine focus and have to back off, use the coarse, then back to the fine. I recommend making sure the microscope has an adjustable iris, as that can be important in looking at some things. I have lots of interested kids, so I went with the binocular+third vertical eyepieces so I can look through one and two kids can also look through the others. I could also attach a camera to the third if I wanted. For the record, I seem incapable of using the binocular eyepieces, but I think that's a personal deficiency -- I also can't see magic eye things and I think it is related. My wife loves looking through the binocular scope (she is also a glasses-wearer and usually just takes them off and uses the microscope focus plus diopter to view things, which means they will be out-of-focus for the third eye, but that's how it will be.)
Next is the lenses, and there are two kinds to worry about: eyepiece and objective. The eyepiece lens is what you look through. In simple terms, I'd say go with 10x eyepieces, and no higher. There is really no benefit to higher magnification eyepieces until the entire optical system is improved for physics reasons. It just blows up the same blurry image past about that point. It's also likely that there is no reason to go with a lower magnification eyepiece (at least as your primary eyepiece). Just keep it simple there, and if you sometime had a compelling reason to use something else, buy that separately and swap it in as needed. For the objective lenses (the ones close to the sample) you mostly need to decide: do you want especially low magnification (if so, make sure you get a 4x objective for 40x magnification) and do you want to look at bacteria and especially small things (if so, make sure you have a 100x oil immersion lens for 1000x magnification). Don't forget that if you only wanted to rarely look at bacteria, for instance, you
could choose to just buy that objective lens separately and swap it in as desired (if that was a better deal and you didn't mind it). I have a microscope with 4 objectives: 4x, 10x, 40x, 100x.
I recommend the oil immersion lens, and I don't think you'll regret having it, and I think you'd prefer it just being there.
Finally, there are the additional things you will need/want. You will want a box of blank slides and coverslips. You need alcohol and wipes to clean them (even if they say they are pre-washed...) and to clean the lenses. You will likely want some kinds of stains, depending on what you're looking at since cells tend to be transparent. Esoin Y is great for plant matter. If you have the oil immersion lens, I expect you'd want to do gram staining which needs crystal violet, iodine and safranin O. There are other stains for living things (paramecia, amoebas) for viewing them swimming around, but I don't have
experience with that (you also need slides with concave depressions to put their drops of
water in). You also need immersion oil if you have that lens (but they might send you a small sample size with the lens). You might also want to get prepared slides which will likely be better than what you can make at home (but much less fun, to be clear!) and permanent (though you can do that at home as well).
In summary, you can get a very nice microscope at the $200-$300 range for home use and you might need to decide what features you want to narrow it down to a final choice. I'll leave you with a couple of shots I took with my phone through the eyepiece of mine to give you an idea of what you can achieve with a simple set-up. The phone was hand-held so the images are not the best, but that's what I've got available right now; I assure you it looks much better with your eyeballs.
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