First mission is typically a procurement mission. Be careful what you are asked for. If it is to get combat stabilizers, just ignore. Those are usually not anywhere close and are illegal in most stations. Wait for non-lethal weapons request to pop up.
There are four types of missions so far. First is a procurement, next couple are a delivery ones, them comes an assassination and last type is to hunt pirates. If you just did a mission, come back to port and there is another one waiting, don't take it - you will not get another progression.