First, I'd start by getting your lens. Seriously--start with the lens. The lens (a good one) will be more expensive than your body. You're probably going to want something like a 25-75mm f2.8. That will be good for portraits, street photography, landscapes, architecture and fast enough for indoor with ambient light. Get a good 25-75mm f2.8 (or something like that: 20-70mm or 27-80mm, f2.8). The f2.8 is important because it produces lovely portraits (by blurring the foreground and background). And it allows you to get good photos even when the sun isn't shining brightly. And then buy a body that fits that.
I know, I know, that sounds bassackwards. But for $300 (or less for your body), your lens will be the most expensive (and critical part) of your shooting gear. Find a lens that will work for your purpose. Finding a body to go with it will be easy, especially since you're looking at "used." Lots of options there.
Also, if you buy from a camera store or someplace like
Adorama or BEH, you can pick out the lens and then say "my budget is $300--what is a good DSLR for that lens?" And you can assemble a package (lens, body, maybe an extra battery, maybe a speed light or other stuff). Make sure you buy a battery charger with your camera.
I will give you a hint: the trend these days is to go mirrorless. That is where the camera companies are going. But mirrorless glass tends to be more expensive. And there are many bargains to be had (especially from people seeking to sell their DSLRs as they plunk down $3,000 or more for a mirrorless kit. I can almost guarantee you shouldn't buy mirrorless (because to get a mirrorless body under $300 means you're probably buying one that is very beat up or very early technology with a mediocre sensor).
A few other tips. Since you're a beginner, plan on investing in a manual or a book on your camera (the book is almost always a better read than the manual). If you want to do video, consider a tripod--they're also useful for landscapes. A good one will cost you a couple of hundred bucks. The cheap ones ($30-$40) aren't very stable for anything beyond a camera phone. If you buy mirrorless (especially a first gen mirrorless) you should buy a couple of extra batteries--DSLRs get about 3x-4x the number of shots per battery as a mirrorless. Older mirrorless cameras are even worse in that regard.
Finally, based on your description you made it sound like you are shooting subjects relatively close by. if you get interested in sports (a kid's club team) or wildlife or speciality topics (like macro photography) you're going to need more than one lens to start out with. You'll see general purposes lens (20mm-120mm or 30mm-300mm) available--but they'll be outside of your price range or very "soft" and poor in low light (meaning--any place indoors where you aren't using a flash or speed light). Do not count on the popup flash on your camera to provide good light--it's best for "fill" and not to light a scene (where it's a picture killer--creating flat photos or faces with lots of glare on them). So if you're going to shoot a lot indoors, consider buying a speed light to add on to your camera (you can buy Asian knockoffs like YongNuo on Amazon for $40) or some LED continuous lights to operate off of your camera.