I need my method to return two Maps. Is this possible?

Apex does not allow multiple return types. However, you can create a custom type to be returned from your method. It would have a Lead and a Set<Date> and a Set<String> in it - allowing you to have strings and dates per Lead.

Similar to this:

public class LeadWrapperThing {
    public Lead theLead             { get; set; }
    public Set<String> theStrings   { get; set; }
    public Set<Date> theDates       { get; set; }
}

public List<LeadWrapperThing> getLeadWrapperThings() {

    List<LeadWrapperThing> thingsList = new List<LeadWrapperThing>();

    for (Lead someLead : listOfLeads) {

        // create the wrapper
        LeadWrapperThing lThing = new LeadWrapperThing();

        // stick the lead in the wrapper
        lThing.theLead = someLead;

        // add stuff to the string set
        lThing.theStrings = new Set<String>();
        lThing.theStrings.add('a value');

        // add stuff to the date set
        lThing.theDates = new Set<Date>();
        lThing.theDates.add(Date.today());

        // add the wrapper object into the list to be returned
        thingsList.add(lThing);
    }

    return thingsList;
}

or as an answer to your comment, if you want to use the lead as the key also and return a map

public Map<Lead, LeadWrapperThing> getLeadWrapperThingsMap() {

    Map<Lead, LeadWrapperThing> thingsMap = new Map<Lead, LeadWrapperThing>();

    for (Lead someLead : listOfLeads) {

        // create the wrapper
        LeadWrapperThing lThing = new LeadWrapperThing();

        // stick the lead in the wrapper
        lThing.theLead = someLead;

        // add stuff to the string set
        lThing.theStrings = new Set<String>();
        lThing.theStrings.add('a value');

        // add stuff to the date set
        lThing.theDates = new Set<Date>();
        lThing.theDates.add(Date.today());

        // add the wrapper object into the Map to be returned, using the lead as the key
        thingsMap.put(someLead, lThing);
    }

    return thingsMap;
}

// Usage:
// Map<Lead, LeadWrapperThing> wrappedLeadMap = yourClass.getLeadWrapperThingsMap();
// LeadWrapperThing wrappedThingForThisLead = wrappedLeadMap.get(theLead);

In addition to creating a new class to hold the values as Mark Pond suggested, you could take advantage of the fact that Maps, like all collection types, are reference types rather than value types, and simply provide variables for the method to fill as parameters.

void fillLeadStringDates(Map<Lead, Set<String>> stringset, Map<Lead, Set<Date>> dateset)
{   
    stringset.put(aNewLead, 'a string');
    dateset.put(aNewLead, Date.newInstance(0, 0, 0));
}