I had language partner in France (the program was called Tandem there) and I feel the whole purpose of such a program is speaking the foreign language. It's especially well-suited for learning daily life phrases and expressions that you don't find in books or films (I think even movies have their "own" language). That's why we often talked about customs of the respective foreign culture. You should aks your partner how to say this and that in certain situations. And it's always great fun to exchange cliches about the foreign country and their people. Here are some suggestions:
Wie feiert man in Deutschland eigentlich Weihnachten?
Wieviel Trinkgeld gibt man in einem Restaurant?
Wo würdest du typisch deutsche Geschenke für meine Freunde in [your home country] kaufen?
Was sage ich, wenn mir jemand auf den Fuß tritt und sich dafür entschuldigt?
Sind die Deutschen eigentlich wirklich so ordentlich wie alle sagen?
Talking about grammar is more difficult and I would not recommend it in this setting. Most native speakers have difficulties to explain their native language in formal terms because it's just natural for them.
Discussing politics and the news is also interesting but requires an advanced level.
Oh, and when it comes to games, I remember one that I loved to play as a child. It's called "Ich sehe 'was, das du nicht siehst". It's for two players. One player randomly picks an object in the surrounding not telling the other player what it is. She/he only reveals the color of the object:
Ich sehe 'was, das du nicht siehst und das ist blau.
Then the other player has to guess what it is:
Die Handtasche der Frau? - Nein.
Die Zeitschrift auf dem Nachbartisch? - Nein.
Mein T-Shirt? - Ja!
Like this you can learn vocabulary of everyday-life and you don't need anything to play it.