I have found that the more regularly you break out a board game with your kids, the more fun it becomes for everyone. It is a chance to play together, talk together, and simply be together without a piece of electronics acting as a barrier between your child and the attention you need to give her that makes her feel loved and understood. That is the kind of attention which buoys up your child through the difficulties she may face from time to time and gives her the confidence in her own abilities and judgement.
I always offer a deal with my kids: one round through a new (perhaps learning-based game) in exchange for a pile of rounds of a game they know and love. Yesterday we traded a few rounds of Scrabble Up (which turned out to be so much fun for them that they played for nearly and hour), for a couple of rounds of Clue and one game of Risk.
Actually my kids learn every time they play any kind of game, but we define learning games usually as those that directly practice or reinforce lessons. The fun games that they get to pick really teach the most important lessons of all: character, compassion, and confidence.
