We have a very bad mold problem in our apartment. What health concerns should I be worried about for myself and my children.?

Two. Some types of mold secrete a respiratory toxin; if anyone in the family quickly develops a cough or asthma-like symptoms, you need to move. Anyone in the family allergic to the molds in the home may develop recurrent sinus infections or the gradual onset of asthma symptoms; skin testing by an allergist can be helpful to sort this out, and possibly prove the problem to the landlord.
Multiple issues. Based on on studies, significant exposure to mold can cause worsening rhinitis and asthma symptoms. It also increases the risk of asthma and allergies in children as well as adults.