Thanks to the above answers I was able to do exactly what I wanted, here's exactly what I did for anyone looking for this in the future:
My activity_main.xml looks like this:
<!--When the DrawerLayout is the root layout, the first child-->
<!--of that layout is the contents of the main screen, and the-->
<!--second child is the contents of the menu-->
<!--First child layout-->
<LinearLayout
    android:layout_width="match_parent"
    android:layout_height="match_parent"
    android:orientation="vertical">
    <include
        layout="@layout/toolbar_layout"
        android:layout_width="match_parent"
        android:layout_height="wrap_content" />
    <FrameLayout
        android:layout_width="match_parent"
        android:layout_height="match_parent"
        android:id="@+id/content_frame"/>
</LinearLayout>
<!--Second child layout-->
<android.support.design.widget.NavigationView
    android:id="@+id/navigation_view"
    android:layout_width="match_parent"
    android:layout_height="wrap_content"
    android:layout_gravity="start"
    app:headerLayout="@layout/navigation_drawer_header"
    app:menu="@menu/drawer_menu">
</android.support.design.widget.NavigationView>
This is the standard DrawerLayout which pulls together all the bits and pieces for the NavigationDrawer menu. The important addition to this is the FrameLayout... bit to which I gave the ID content_frame. This is where all the other layouts used by other activities will be pushed/added/inflated.
My BaseActivity.java looks like this:
package com.example.test;
import android.content.Intent;
import android.os.Bundle;
import android.support.design.widget.NavigationView;
import android.support.v4.widget.DrawerLayout;
import android.support.v7.app.ActionBarDrawerToggle;
import android.support.v7.app.AppCompatActivity;
import android.support.v7.widget.Toolbar;
import android.view.MenuItem;
public class BaseActivity extends AppCompatActivity {
    DrawerLayout drawerLayout;
    ActionBarDrawerToggle actionBarDrawerToggle;
    Toolbar toolbar;
    @Override
    protected void onCreate(Bundle savedInstanceState) {
        super.onCreate(savedInstanceState);
        setContentView(R.layout.activity_main);
        NavigationView navigationView = (NavigationView) findViewById(R.id.navigation_view);
        toolbar = (Toolbar) findViewById(R.id.toolbar);
        setSupportActionBar(toolbar);
        drawerLayout = (DrawerLayout) findViewById(R.id.drawer_layout);
        actionBarDrawerToggle = new ActionBarDrawerToggle(this, drawerLayout, toolbar, R.string.drawer_open, R.string.drawer_closed);
        drawerLayout.setDrawerListener(actionBarDrawerToggle);
        navigationView.setNavigationItemSelectedListener(new NavigationView.OnNavigationItemSelectedListener() {
            @Override
            public boolean onNavigationItemSelected(MenuItem item) {
                switch (item.getItemId()) {
                    case R.id.menu_home:
                        Intent anIntent = new Intent(getApplicationContext(), TheClassYouWantToLoad.class);
                        startActivity(loadPlayer);
                        drawerLayout.closeDrawers();
                        break;
                }
                return false;
            }
        });
    }
    @Override
    protected void onPostCreate(Bundle savedInstanceState) {
        super.onPostCreate(savedInstanceState);
        actionBarDrawerToggle.syncState();
    }
}
Now, within the onNavigationItemSelected method, there is a switch statement that handles what happens when each menu item is selected. This is the important bit:
Intent anIntent = new Intent(getApplicationContext(), TheClassYouWantToLoad.class);
startActivity(anIntent);
drawerLayout.closeDrawers(); 
You need to replace "TheClassYouWantToLoad" with your own class.
Now within this new class (which presumably requires some new UI elements to be loaded), you need the following:
public class TheClassYouWantToLoad extends BaseActivity {
@Override
protected void onCreate(Bundle savedInstanceState) {
super.onCreate(savedInstanceState);
FrameLayout contentFrameLayout = (FrameLayout) findViewById(R.id.content_frame); //Remember this is the FrameLayout area within your activity_main.xml
getLayoutInflater().inflate(R.layout.the_layout_you_want_to_load, contentFrameLayout);
    }
}
and replace "the_layout_you_want_to_load" with the name of the Layout you want to load.