trigger Send_Qtrly_Update_02_02 on Task(after insert, after update) {
    // next call date is 90 days out
    Date nextCallDate = DateTime.now().addDays(90).date();
    
    // Tasks that meet criteria, and new tasks to create
    Task[] qualifiedTasks = new Task[0], newTasks = new Task[0];
    
    // Map of provider facilities
    Map<Id,Provider_Facility__c> facilities = new Map<Id,Provider_Facility__c>();

    // Find qualifying tasks
    for(Task record:Trigger.new)
        if(record.subject=='Quarterly Outreach Call' && t.isclosed && record.whatid!=null && record.whatid.getsobjecttype()==provider_facility__c.sobjecttype)
            qualifiedtasks.add(record);
    
    // Obtain facility ID values
    for(Task record:qualifiedtasks)
        facilities.put(record.whatid,null);
    
    // If there are any facilities to query, do so.
    if(!facilities.isempty())
        facilities.putall([select id,outreach_agent__c from provider_facility__c where id in :facilities.keyset()]);
    
    // For each qualifying task, create a new task and update the facility record.
    for(Task record:qualifiedtasks) {
        newTasks.add(new task(whatid=record.whatid,subject='Quarterly Outreach Call',ActivityDate=nextCallDate,OwnerId=facilities.get(record.whatid).Outreach_Agent__c));
        facilities.get(record.whatid).last_outreach_call_made__c=Date.today();
    }
    
    // Insert new task records into database.
    insert newTasks;
    // Update facility last call dates.
    update facilities.values();
}

This encompasses all of the best practices I could think of:

* Use a map instead of querying inside a loop.

* Don't use extra variables if you can avoid it.

* Prefilter Trigger.new to find records you will be operating on.

This code is basically what I call the "aggregate-query-update" pattern. First, I gather together all the values I need, then I query for them, and then I perform updates based on the queried data. This is the most common form of a trigger, so learning this pattern will make sure your code is always bulkified.