You can make it fun. Role playing for example. You can send your partner a couple of scenarios you would like to act out in the language you are learning. I am thinking about doing something like this. 

You can also try word games. Like charades or twenty guesses.

I can advise you to digress for the first conversation, partly because it creates a friendly fun relationship and it's less stressing.If you go into a language exchange relationship with a full on "I'm here for business" attitude it is extremely stressful on both parties to keep up a very formal and serious conversation with someone you barely know. I digress partly because I like talking a lot and partly because of the audience.

I am speaking to another human being, usually my age, therefore I act accordingly, yet of course I come back to the original topic. If you have a language partner you can talk about pretty much anything that is culturally different. I for example, talk about a lot of school related things with my fellow partners, "Oh so you stay with your same classmates for every class? Does it get boring to be with the same people or does it mean you guys are closer?"(me speaking to a Korean language partner).

To sum it up, treat your partner, like a partner, they are your buddies, create a relaxed relationship so you can discuss any and all topics you are curious about, especially the culturally significant topics.

Thanks for the suggestion Mattewtrujillo! :)

Ana,

Thanks for your comment. I do agree that I would love to get off subject and talk spontaneously with my language partners. However sometimes it is hard to find a partner like that who can maintain such a free conversation. 

I have had some conversations and in some of them it was very hard to keep a flowing dialogue. (Of course, I mean after we had talked about our lives, work/school and all those other beginner topics that you cover within the first 3 to 4 meetings.)

I am actually asking for some new topics to try that will help expanded the conversation and keep it interesting. Topics that will be useful for a conversation partner you have had for a while, not just a new partner that you are just starting out with.

But thanks for you're example and response.