Ah OP I get the anxiety but I promise you, London is really easy to navigate, everything is so well signposted, loud clear announcements everywhere and (contrary to popular opinion) most Londoners very friendly and all too happy to help visitors especially parents with kids (just please don't stand on the wrong side of the escalators or block the platform entrances!).
Either green park or Hyde park corner are on the Piccadilly line (dark blue), as soon as you get off the train head towards the ticket barriers and you should fairly quickly spot signs directing you to the underground (make it a game for your DC, who can spot the tube logo first, who can spot Piccadilly line first etc). The staff at the barriers will point you in the right direction if you're not sure. If at any point you need to regroup, faff with a buggy or gather your children, do so but find a spot to the side, like I say Londoners are perfectly nice but there's a flow of foot traffic around busy stations and you will cause a jam if you stop dead in a high traffic zone. IIRC correctly at Kings Cross it's a fairly long walk down to the tube platforms, probably multiple sets of stairs and escalators (there are lifts too if needed but again probably more than one), and there maybe more than one entrance so don't be confused if you see that. There are also lots of intersecting lines at kings cross (circle/Hammersmith and city/victoria/northern etc) as well as Piccadilly so once into the underground station there will be a , but just keep looking out for the dark blue. Use the big luggage gates at the tube barriers (do NOT try and squeeze your DC through the regular barriers!), kids under 12 are free and you can use your regular contactless card or gpay/apple pay on your phone to tap in and out. Again staff will help and direct you as needed.
At kings cross I am sure the Piccadilly platforms are clearly marked eastbound and westbound to a PP that mentioned that they aren't always, you want westbound. I'm afraid I can't say you need a train to 'x' destination as westbound Piccadilly trains go to a number of different places, Heathrow terminals, Uxbridge etc but for your purposes so long as you're on the westbound platform any will do, so don't worry about the final destination shown on the board. I think green park is 4 stops from kings cross and Hyde park corner 7.
There are big signs everywhere showing all the stops as well as whole tube maps, on the platform and on the train and every stop is clearly announces as upcoming. Again your DC might enjoy working out your journey, counting the stops etc, you could even pick them up a paper tube map each for free at the station and they can follow along, mine bizarrely used to love calculating the most efficient journeys across London to and from obscure suburbs 
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In the middle of the day it shouldn't be too busy on the tube but teach your DC good tube etiquette anyway, stand behind the yellow line, wait for people to leave the tube before getting on, never barge or push, if one train is too full to get on don't panic, there'll be another less than 5 mins behind. People will likely give your DC a seat (ask if none free) but they might prefer to stand, make sure they hold on though! Basic tube safety/rule of thumb in the hugely unlikely possibility of someone getting separated from their party is get off at the next stop and wait there/ find a member of staff to help.
The parks are well signposted from the tube, look for the black and yellow street maps while out and about too.
On the way back you want an eastbound train which will all go to cockfosters. Citymapper is a great app so I'd get that or google maps also v useful for directions and also searching for cafes, restaurants, toilets, ATMs, anything you might need really!
Good luck and enjoy!