Which lens do you recommend for entry level bird and wildlife photography on Canon APS-C DSLRs?

Ignoring the specific lenses here, the question you're really asking is "which is better, a general purpose 'superzoom' lens or a dedicated telephoto lens?" The answer to that is of course "they're different":

• The superzoom lens (18-200 in your case) has the advantage that it has a very wide focal length range; it can do everything from landscape photos to some wildlife photos (see below for more on that). However, nothing in this world comes for free - the image quality won't be as good as dedicated lenses, and you pay a premium for flexibility.
• The dedicated telephoto lens will probably have better image quality for the range that it covers, and is cheaper.

Only you (or your son, or whoever's paying) can decide which of those tradeoffs is right for you. Three notes though:

1. You lose a lot of the value of an interchangeable lens camera if you only ever use one lens. Unless you're in a sandstorm or under a waterfall, changing lenses isn't a risky process so long as you're moderately careful.
2. While we in general don't recommend specific products, I'm prepared to break that rule for the 75-300. Never, ever buy this lens - it's image quality is awful and it doesn't have image stabilisation. For a crop sensor camera like the 80D, you're much better off getting the 55-250 and cropping the image.
3. Even a 300mm probably isn't going to let your son get good images of wildlife unless the wildlife is pretty tame; for bird photos in particular, you'll be looking at 500mm+, but those lenses are much more expensive. Just setting expectations here.

Straight up, I should mention that I am not a bird photographer and do not have much experience in this area. Hopefully someone with more experience can offer an answer with more specifics.

Preferred Zoom Range

However, the bird photographers that I know all seem to gravitate toward use of 150-600mm zoom lenses (that ... and techniques to allow them to get closer to shy birds without scaring them off.)

Even lenses that zoom up to 300mm seem to not be enough. Canon makes a 100-400 and even that doesn't seem to be enough. Use of a teleconverter is not desirable as it creates focus issues.

Auto-focus and Focal Ratios

The focus technology used in the cameras (when using the optical viewfinder) relies on passing the light through a beam splitter and detecting the phase-difference of the light. But this system only works well at low focal ratios (e.g. f/5.6 or lower ... even f/6.3 is fine for most cameras). At f/8, many cameras are no longer able to reliably focus ... but a few models can operate at f/8. I am not aware of any camera that can use phase-detect auto-focus above f/8. And this is the problem with teleconverters used with long lenses.

A 70-300mm zoom will typically have a variable focal ratio of f/4-5.6. At the 300mm end, it's lowest focal ratio is f/5.6 (f/4 is only available near the 70mm end of the zoom range). When you add a teleconverter, you multiply the focal ratio of the lens by the focal length multiplier (e.g. if it's a 2x teleconverter then you multiply by 2). This means 5.6 becomes 11.2 (f/11) and no camera can use phase-detect autofocus at f/11.

A Canon 100-400 f/4-5.6 lens coupled with a 1.4x teleconverter effectively becomes a 140-560mm lens and at 560mm it's lowest focal ratio is f/8 ... and some cameras can still use some of the auto-focus points at f/8 (but expect auto-focus speeds to be slower). So this solution can work but probably isn't ideal.

And I think this is the reason why most bird photographers tend to prefer the lenses that natively have the 150-600mm range (and these are usually f/4.5-6.3 but the camera can auto-focus at f/6.3).

Lenses

Your budget will factor into your choice but here are a few options:

Sigma 150-600mm F5-6.3 DG OS HSM | S -- this is Sigma's top-end 150-600mm zoom. The final letter "S" is "Sport". Unsurprisingly it is also their most expensive lens offered in this range.

Sigma 150-600mm F5-6.3 DG OS HSM | C -- this is Sigma's less-expensive offering. The final letter "C" Is "Contemporary". This lens sells for about half the price of the "Sport" version.

Tamron SP 150-600mm F5-6.3 Di VC USD G2 -- the "G2" version is Tamron's newest (generation 2) version of the lens and it is the more expensive model.

Tamron SP 150-600mm F5-6.3 Di VC USD -- the non-G2 version is the older design for Tamron but they still make and sell this lens. It is significantly less expensive than their newest version.

Sigma and Tamron make these lenses with mounts for a variety of manufacturers (e.g. Canon, Nikon, Sony, etc.) so if you do choose one of these, make sure you order the lens in the Canon-mount model.

Again, I am only offering that these are the lenses used by most of the bird photographers that I have met. I have no personal experience with any of these lenses ... but offer this list as a starting point in your search.

Your question is whether you should go for "a 75-300mm zoom lens or a 18-200mm zoom lens." You have short-listed these two kinds, based on your budget, I presume. Therefore, without describing the advantages of going for other more expensive or specialized lenses, let me give you a straight-forward and simple answer. [When someone asks for a drink, there's no point overwhelming him with a jet from a fire-hydrant!]

Go for the 75-300mm zoom lens.

Yes indeed, the 18-200mm lens can serve as a "multi-purpose' lens. However, since you mentioned bird photography, I suggest you go for the 75-300mm lens. While shooting birds, your son can focus on that task and then, after he's done with it, conveniently change to his kit lens to take wide-angle shots. He will always appreciate the additional zoom available in the 75-300mm lens.

HINT: Along with a zoom lens, it's always advisable to use a tripod or at least a mono-pod.

I can't recommend the EF 75-300mm f/4-5.6 III to anyone. It's probably the weakest lens in Canon's entire catalog. It's at the softest at the longest focal length of 300mm.

A wide zoom range lens like the 18-200mm also comes with compromises in optical quality and maximum aperture in order to allow it to have such a wide focal length range. An 18-55mm + 55-250mm set will outperform the 18-200mm in pretty much every measure other than not having to change lenses.

Neither the 75-300mm nor the 18-250mm would be very good for your stated purpose.

At the lower end of the budget range, the Canon EF-S 55-250mm f/4-5.6 IS STM is probably your best choice. It performs better optically than either of the lenses you are looking at and is priced a little less than half what the 18-200mm currently costs. On your son's 800D, the EF-S 55-250mm f/4-5.6 IS STM is about as good optically as the higher priced EF 70-300mm f/4-5.6 IS II. The main difference, other than 55-250mm vs. 70-300mm, is that the 70-300mm lens projects a larger image circle and can be used on Full Frame cameras with sensors larger than the APS-C sensor in the 800D. The EF-S 55-250mm lenses are limited to use with "crop body" cameras with APS-C sensors like your son's 800D.

For not much more than the current $700 cost of the EF 18-200mm f/3.5-5.6 IS, you can get one of the 150-600mm offerings from Tamron and Sigma. Sigma is currently running a pre-Black Friday promotion and dealers are selling the Sigma 150-600mm f/5-6.3 DG OS HSM C for around $900 USD in the U.S. The very similar Tamron 150-600mm f/5-6.3 Di VC USD G2 is currently selling for around $200 more. Both prices are about 10% lower than their recent normal retail cost.

For a beginner, the EF-S 55-250mm f/4-5.6 IS STM or an EF 70-300mm f/4-5.6 IS II will be a little easier to learn to use than one of the 150-600mm lenses. It will perhaps also be more useful for a variety of other purposes that the 18-55mm lens is too short to do well and one of the 150-600mm lenses are too long to be able to do. The longer focal length one uses, the better one's shooting technique needs to be to avoid allowing camera movement during the exposure to cause noticeable image blur. Image Stabilization does help deal with camera movement, but better technique combined with IS will go even further than poor technique with IS will. It's also harder to keep a moving target in the viewfinder with a longer focal length telephoto lens than it is with a shorter telephoto lens.

On the other hand, when used well a 150-600mm lens will probably get the kind of results desired by those who shoot wildlife and birds in flight than a 55-250mm lens will.

Following up on my bird feeder comment I took some quick shots to show roughly what this looks like. Our feeder is 20 feet (6m) away from where I would shoot from. I find I prefer shots where the birds are in the trees on the way in to the feeder or the way out. The photo was shot at 200mm on an APS-C sensor from about 25 feet (8m). It is a chestnut-backed chickadee, a normal to small feeder bird at 4.75 inches (12 cm) long. I have adjusted the light in Lightroom but nothing else. The light was dim, so I used ISO 1600, f/9, 1/100 second. It is rather grainy, but I left that. It shows what image size you can expect from a lens of this focal length.