We have enthusiastically played Candy Land with each of our three kids.  It is easily the best starting point for introducing a child to board games.  Players take turns drawing from a deck and moving their pieces around the board.  The cards have one or two squares of a particular color (such as red, green, yellow, etc.) and players move pieces ahead on the path to the next matching spot of the same color.  No counting or reading is required.  There are also special cards that can move a player ahead or behind, and they are indicated by  picture of the spot on the board to which the player should move their piece.  Play continues until one player reaches the end.