What will the cooking and cleaning roles be?
You can determine who is responsible for what cleaning duties by creating a list. For example I prefer not to clean the shower so I guarantee I will vacuum and sweep instead. Trade offs and negotiations are important. Try and set a schedule, even if it is a loose schedule so there is no argument about when a cleaning task needs to be done.
Who will be the chef?
This can change depending on schedules or interest. I recommend deciding what you will eat at least one to two days in advance and from there you can choose if you will cook together or if one of you will go it alone. If that is the case, the other should wash the dishes. When in need of simple recipes Trade Joe’s is a great resource.