Once your kiddo has learned to take turns and not touch the game pieces you can move on to more classic board games. One step closer to family game night!
I would recommend starting with games that have a spinners, dice and moving the pieces. Avoid games that have too many rules.
Tip: To avoid having your kid take over the board game and move pieces all over the board (and make you want to just give up), make the rule that only the adult touches the board and moves the pieces!