This is an important question. I cannot give a complete answer, but I will try to give a partial one.
I will try to edit and add to this post whenever I have new input - even though this might be years from now.
Doing the math is one of the easiest techniques of fact-checking of any number-based claim. You rarely have exact numbers, but doing a Fermi-style estimation ("how many piano tunes exist in Chicago?") is often enough.
Children who rely on a calculator todo multiplications will never be able to do quick estimations in their head.
Luckily, learning multiplication tables can be a game, and if you have multiple children you can do a competition: Ten multiplication-questions in a row, the winner is who answerd first and correct to most questions.
"I bet this tree has 100.000 leafes!"
Do a game called "I bet this tree has 100.000 leafes!". You make a bold estimation of something. Your kids can either answer with "That could be true!", "Many Many More!" or "Far, Far, Fewer!".
Then, together, you check the estimation: how many leaves has a twig, how many twigs a small branch, how many small branches in a big one, and so on.
(You see how the multiplication tables come in useful!)
Make sure you explain rounding early on and apply it after each step. Otherwise the calculations can get frustrating to your kids (and yourself) pretty soon.
You and anybody who agreed with you win if you got the numbers (the magnitude) roughly right. If it is more than 10-times more or less, it is a win for "Many Many More!" or "Far Far Fewer!".
If you connect this game with any price, use something non-imediate. If the winner wants their chocolade-bar, they are to impatiant to double-check or discuss the result. But if the winner gets to choose the desert of today's dinner, you will have all afternoon to write things on paper, and discuss with your kids (if they are interested - do not force them ;-) ).
Introducing volumes and units
After playing the game a few times (one bet each time you play), teach how to convert units (including common rough estimations)and how to calculate areas and volumes.
The question then might be: "I bet this swimming-pool has 1 Million liters of water!". Which includes calculating the volume based on side length and deducing liters from cubic-meters.
What use is it?
There are claims that try to shock you with numbers. Example: "To produce car-batteries, 25 million liters of groundwater are pumped up in South America every day."
Instead of blindly sharing, your kids might do a quick calculation. (They might find that it is believable, but that is only the contents of a few swimming-pools - hardly a problem considering South-America is a continent.)
Numbers of the world
How many pupils has their school, how many people live in your town? How many people live in your state and nation? How many people are on each continent?
Political believes, mindsets
Understanding the values and believes of (opposing?) groups can help estimating if an accusation makes sense. Before telling your children who you think is wrong or right, give them a rather neutral description in what each party and significant group in your area (and on the news) believes in. What is important to them? Why do they believe to be "the good ones"? Again: Do make this as neutral as possible.
Also explain common words like "conservative", "progressive", "liberal", "social(ist)", "communist", "left", "right". Do mention that the meaning of these words can differ vastly in different cultures (What Germans consider their conservative party would be far left in the spectrum of the US, for example).
If you have trouble speaking neutral about some current-day believes, at least explain in which context they developed and why people believe(d) in them.
Trustworthyness of Media
Game: "Calm or Clickbait?"
You should explain at some point the various aspects that influence the content of newspaper, television and other media. Certainly you should mention "appeasing to the audience" and why some news-sources bend the truth to get more "spectacle-like" titles and content.
You can train your kids in paying attention to "click-bait-titles" (even in non-electronic media). Point to headlines of newspaper in the super-market and ask "Calm or Clickbait?" and let your kids discuss. Do the same for television-news ("A rackoon made the evening news? Is this calm or clickbait?").
At some point turn the game around: Ask the kids to do a clickbait-title for some boring(?) family-event. They are allowed to bend the truth, exagerate and leave out details, but they are not allowed to outright lie.
Soon they will be far more sensitive in believing something they recognize as click-bait.
"who said it?"
When I grew up, my only source was the local newspaper. Articles often began with "A scientific study showed..."
No hint of who did the study, how valid is was, who payed for it and where to find the full source. We had to believe in what the newspaper stated, even if the reporter missunderstood or even twisted the content of the study.
Today we live in an information-paradice where most of scientific studies can be found online.
So do challenge your kids when they find some wild claim: "That feels odd to me. Can you find the original source?"
Most of these sources will be to dry for your kids to read. But you can read them and point out discrepancies to what was claimed: Were the questions in a survey twisted to begin with? Was the study cited in an unfair or biased way? Does it apply to the context? What is the sample-size?
Often, your kids might find that a source can not actually be found - which is also ok, since it gives them some healthy doubt about what they read.
This excersise will also train their online-research skills. Also it will teach them that a statement does not become true if it is repeated often enough, a statement has to be based on something.