List It Up
Use the alphabet to come up with a list of things about yourself while your partner does the same. Take turns guessing what the other put for each letter and talk about your answers. Remember this is supposed to be fun and get you talking, not to turn into an argument, so if there is an answer you do not like, relax and just go with it.
Grab a bowl and fill it with one-word ideas that the two of you came up with and write on scrap paper. These ideas can be anything you want in one word, from pizza to body parts and anything in between. Take turns picking ideas from the bowl and say the first thing that comes to mind when you read it.
Everyone has a favorite food, color, place or movie, so use these to create a conversation game. Go down the line sharing your favorite foods or restaurants and what you like most about them; follow that with your favorite movies, and see how many you have in common. Next thing you know, your calendar is marked for a date night, dinner and a movie.
This is about as old as it get when it comes to conversation games, but it can be a blast. Begin with something as simple as where you are from, and then your partner just says the first thing that comes to mind. This can go absolutely anywhere and usually does. A conversation about what sports you played as a kid or if you had any pets growing up can come from something as crazy as "macaroni and cheese." Make the first word something personal if that is the direction you want the conversation to take.