Card game for children
Here are my suggestions of games that teach various card game rules, ranked in order from least difficult to most difficult.

Go Fish has simple rules and can help you learn basics of card games, such as


Identifying matching sets of cards
Identifying card names and suits
Waiting your turn
Communicating with opponents when required (and being truthful about it)
Drawing and holding cards without revealing them


War kicks the
competitive level up a bit and has more rules. In addition, you learn card value progression, including the hierarchy of J-K (which can be explained fairly simply if the player is familiar with monarchy, fictional or otherwise).


Competition
Card values
Winning a round without winning the game


Bullshit is a matching game, like Go Fish, but it adds the concept of a bluff and shows that following the rules can sometimes feel like cheating, since duping your opponent is truly the objective of the game. This is useful when later learning games like poker. Also, this can introduce the concept of 'house rules', and hones your ability to agree on extra rules once the game is in progress and tensions are high. Requires 3 or more players.

Gin Rummy has quite a few rules in theory, but is quite simple to play most of the time. It provides practice for:


Identification of runs
Using the discard pile to draw from
Keeping score