Is there a good site for taking adults from learning how the pieces move to being able to play at a 300+ level?
I don’t know any websites offhand specifically for adult beginners.
If your willing to buy a chess program that is designed for learning, then I’d suggest Chess King Training.
-Save your money and get the cheapest version. I’ve worked my way up to 1850 setting, and the main engine, Houdini 2, doesn’t kick in until I get to 2000.
Needless to say, I doubt Magnus Carlsen could be Houdini 2 at it’s highest setting.
It’s main “classical game” is set up as a ladder, so it starts off blundering really bad and slowly gets harder.
-Just for reference, at 1850, it still blunders, but it can be rather tricky to hold on to that advantage.
-For reference, it’s taken me several months to work up to 1850, having started out at around 1250. No relation to a USCF rating, but as you work up, you’ll definitely see yourself improving.
At the lower levels, it will blunder left and right. 
If your looking for more training material, the website also offers a ton of training modules using the Peshka interface. The only one I’d recommend for a beginner is the Training Package for Beginners. It will easily take you over a 1000 USCF rating, and probably more like 1200 without too much trouble.
You should try and get a handle on some opening theory. Maybe someone else here can suggest a beginners book on openings. Something that explains the first few moves. Most opening books in my experience whip through the first few moves, then start on variations after that.
Some people don’t care for Chess King Training, but I think if you got some peshka modules for beginners to supplement the Chess King Training, like I one I mentioned, it should be enough material to easily take your studies quite far.
I wanted to mention one thing about the Peshka modules, some of the module, especially the mate-in-one puzzles gives you way more problems to do than you actually need to do. On those modules, I’d recommend just doing enough in each section where you feel your making progress or mastering the subject matter.
Actually the site chessvideos.tv would be good for adult beginners. It has all the openings, with the first few moves given for each to show what can be done. I will admit that that is pretty bare bones for those accessing the site for free. Also, it has problems [daily and archive of daily] that can be done as well for free.
Larry Cohen
Why would an adult beginner use a website to learn how to play? There are plenty of 8 year olds who can show you how to play chess at a higher level than 300. They can also show you how to set your chess clock and turn off your phone before a tournament round starts. After you lose to one of them, they will pat your hand and say it is okay to lose a game. They will tell you that learning chess takes time.
I wonder what, if anything, distinguishes a 300 level player from someone who “just knows how to move the pieces”. Most adults will achieve a provisional rating higher than that even if they lose every game.
We need to encourage a raw beginner to just learn the basics and not worry about his USCF rating goals. I’m sure with a few weeks of study with a decent program like Chess King Training, he’d come out of his first tournament with at least a 1000 USCF rating if not more.
-The only reason I plug Chess King Training is because it’s really the only program I found that took the place of the Chessmaster series, which has been discontinued for many years now. I like it a lot. It’s not the same as, say, the Fritz interface [insert chess engine]. Chess King Training is much more educational and it’s two engines are hard coded into it. That is, you can’t load another chess engine into it.
That being said, it’s primary purpose is to help players improve their play.
The two engine are called Robot and Houdini. I chose Houdini 2, but you could get it with Houdini 6 or Houdini 6 pro. Robot is used for sub 2000 play and Houdini for 2000 to 3400+ play.
Also, I wanted to apologize. I was posting I was playing Chess King Training at 1850 and it was still making blunders. It was set at 1750. Apparently I’d changed it back to 1750 and forgot about it. It does make a fairly obvious blunder at 1750, although sometimes I still have to pay attention to find it, sometimes it’s blatantly obvious. I can win at 1850 and I permanently set it at 1850 now. It plays very human like at that level and doesn’t do any obvious blunders anymore. I really have to sit and out-think it to win. I’m winning less than 1 out of 4 games at the moment at 1850, but since it feels more human like, I just decided to play longer games so I can have more time to think.
It’s difficult to dumb down modern chess engines in a way that feels human like until you get close to it’s rating ceiling. Some engines do it better than others. Robot is terrible at faking human play until it gets to about 1800. It’s ceiling is, I guess, is close to 1950. Probably not an easy feat in the era where even a brute force engine could calculate 6 ply in moments and over 10 ply fairly quickly.
I anticipate this married couple in their 60s will be primarily just playing each other to pass the time and will never enter tournaments or spend much time studying.
I assume Tom is speaking tongue in cheek here (at least I thought it was funny). Nowadays when I don’t know how to use something, or something is broken, one of the first things I do is try to find instructions on the web. Most of the time, it’s something that my kids could tell me, but where are they? So I suppose that people will often look on the web to find out something about chess strategy. I typed “How to play chess” to google just now, and I got a list of sites, which look interesting, but of course I have no idea how good or bad any of them are.