How to display picklist in VF and save selected value?

I have a custom object called Program__c which has a picklist called Categories__c.
I would like to make a dropdown list on my VF page that merely lists the contents of the picklist.

My VF page has

<apex:selectList value="{!category}" size="1" multiselect="false" styleClass="pickList" required="true">
    <apex:selectOptions value="{!categoryList}"/>&nbsp;
    <apex:actionSupport event="onchange" action="{!saveCategoryValue}" />
</apex:selectList>

What is selected will be written to another custom object along with other items. Currently, saveCategoryValue doesn’t do anything. I figured I needed to make the list show up before worrying about saving the selected value.

My controller has

public List<SelectOption> getcategoryList()
{ 
    Schema.DescribeFieldResult statusFieldCategory = Program__c.Categories__c.getDescribe(); // good
    List<Program__c> categories = [select id, Categories__c from Program__c];  // get the values from the Program__c object
    List<SelectOption> categoryList= new List<SelectOption> (); // create the list    
    for (Schema.Picklistentry picklistEntry : statusFieldCategory.getPicklistValues())
    {
    // put picklist values into an <apex:selectOptions> tag for VF display
    categoryList.add(new SelectOption(pickListEntry.getValue(),pickListEntry.getLabel()));
    }
    return categoryList;
}

When I preview the VF the drop down is empty.

At the top I have category declared as a string and I think somehow I need to put the selected value in category so that I can save it. Before that I need to see the values.

I’m not understanding this, obviously. What is wrong here?

Thanks for you help!