Since the camera is a SONY a7III...
You DEFINITELY do not want a Canon RF 24-70mm f/2.8 lens. That won't even fit and AFAIK isn't even adaptable for use on the Sony. (Some Canon lenses... EF and EF-S... can be adapted for use on Sony cameras.)
For what you want to shoot (kids, pets, birds), you will most likely need two zooms:
1. A general purpose "walk-around" zoom... a 24-105mm f/4 is a versatile choice that's not too big and heavy. Sony's costs $1195. Depending upon the second lens you buy, an alternative might be Sony's 24-70mm f/4 that sells for about $800. These moderately wide angle to short telephoto f/4 zooms will be reasonably compact and manageable, good for portraits, close ups, interiors, scenic shots and just general purpose. There are also fairly affordable Tamron and Sigma 24-70mm f/2.8 lenses, as well as Sony's own rather pricey 24-70mm f/2.8... but all these f/2.8 lenses will necessarily be bigger and heavier. Or, a variable aperture lens like the Sony 28-70mm f/3.5-5.6 can be smaller and lighter, as well as less expensive.
Here is a pretty comprehensive list of the "walk around" zooms that are compatible with your a7III:
https://www.bhphotovideo.com/c/search?labelId=48194&fct=fct_zooms-primes_5903%7czoom-lenses%2bfct_lens-type_3446%7cstandard-zoom&is=REG&from=detail&showAllAcc=Y&sku=1444401&Note: If you have not purchased the camera yet, often there are favorable deals that include one of these lenses bungled "in kit" with the camera. At B&H Photo, I see the a7III being offered with choice of four different "kit" lenses, as well as "body only" without any lens.
2. You'll want a telephoto zoom for those pets and bird photos, as well as kids playing at a distance and other purposes. The bare minimum you'll want is a 70-200mm or 70-300mm. These can be fine for kids and pets, but pretty short for birding. In fact, birders will sometimes say, "There's no such thing as a 'long enough' telephoto lens!" If birding is important to you, start looking at longer focal lengths such as 100-400mm, a 150-600mm or the recently announced and not-too-terribly-expensive Sony 200-600mm... under $2000, which actually is a really good deal. Once you start buying lenses longer than 300mm, the prices can be as much as a used car! These are some more affordable options, mostly under $2000 and some even under $1000. All these lenses will be bigger and heavier than that "walk around" zoom above. It's just the nature of the beast! The longer the focal length, the larger the lens has to be!
Here's a pretty comprehensive list of telephoto zooms compatible with your camera:
https://www.bhphotovideo.com/c/search?labelId=48194&fct=fct_zooms-primes_5903%7czoom-lenses%2bfct_lens-type_3446%7ctelephoto-zoom&is=REG&from=detail&showAllAcc=Y&sku=1444401&Some might recommend a single "all-in-one" zoom. I usually don't recommend those for several reasons. First, there's actually only one offered specifically for Sony a7III... Sony's own 24-240mm that sells for a bit under $1000. There are other zooms of this type that are "adaptable" for use on the camera, but that's adding the cost of an adapter (about $250), complexity and may or may not give satisfactory performance. Another problem is that all-in-one zooms very typically compromise in some performance factors... AF speed? Image quality? All-in-one zooms are mostly about convenience, with performance factors secondary. Also, when you opt for an all-in-one, you're stuck with a rather large lens all the time. If the all-in-one is the only lens you've got, you can't switch to a smaller, lighter "walk-around" lens when you don't need the telephoto range. Besides, the whole idea of buying an interchangeable lens camera (ILC) is to be able to change lenses in order to adapt it for different purposes. If you're not going to do that, maybe you should be looking at a fixed-lens compact camera instead... maybe the ILC is a waste of money.
An exception is that sometimes when traveling an all-in-one zoom may be useful keeping things to a minimum.... In some situations you may not have the option of carrying extra lenses or changing out lenses at all.
Finally, a 24-240mm on a full frame a7III really isn't "long enough" for birding, if that's important (or for much small wildlife, as well as many sports). The longest all-in-one for other systems (that
might be adaptable for use on an a7III) only go to 300mm, not very much better.
Try to narrow it down to a few appealing lenses, based upon whatever criteria are most important for you (price, intended use, etc.).... then carefully compare things like size, weight, etc.