How can I get continuous location updates in Android like in Google Maps?

Use Fused location provider in Android set your interval in that:

For an example create your activity like this:

public class LocationActivity extends Activity implements
        LocationListener,
        GoogleApiClient.ConnectionCallbacks,
        GoogleApiClient.OnConnectionFailedListener {

    private static final String TAG = "LocationActivity";
    private static final long INTERVAL = 1000 * 10;
    private static final long FASTEST_INTERVAL = 1000 * 5;
    Button btnFusedLocation;
    TextView tvLocation;
    LocationRequest mLocationRequest;
    GoogleApiClient mGoogleApiClient;
    Location mCurrentLocation;
    String mLastUpdateTime;

    protected void createLocationRequest() {
        mLocationRequest = new LocationRequest();
        mLocationRequest.setInterval(INTERVAL);
        mLocationRequest.setFastestInterval(FASTEST_INTERVAL);
        mLocationRequest.setPriority(LocationRequest.PRIORITY_HIGH_ACCURACY);
    }

    @Override
    protected void onCreate(Bundle savedInstanceState) {
        super.onCreate(savedInstanceState);
        Log.d(TAG, "onCreate ...............................");
        //show error dialog if GoolglePlayServices not available
        if (!isGooglePlayServicesAvailable()) {
            finish();
        }
        createLocationRequest();
        mGoogleApiClient = new GoogleApiClient.Builder(this)
                .addApi(LocationServices.API)
                .addConnectionCallbacks(this)
                .addOnConnectionFailedListener(this)
                .build();

        setContentView(R.layout.activity_main);
        tvLocation = (TextView) findViewById(R.id.tvLocation);

        btnFusedLocation = (Button) findViewById(R.id.btnShowLocation);
        btnFusedLocation.setOnClickListener(new View.OnClickListener() {
            @Override
            public void onClick(View arg0) {
                updateUI();
            }
        });

    }

    @Override
    public void onStart() {
        super.onStart();
        if (mGoogleApiClient.isConnected()) {
            startLocationUpdates();
            Log.d(TAG, "Location update resumed .....................");
        }
    }

    @Override
    public void onStop() {
        super.onStop();
        Log.d(TAG, "onStop fired ..............");
        mGoogleApiClient.disconnect();
        Log.d(TAG, "isConnected ...............: " + mGoogleApiClient.isConnected());
    }

    private boolean isGooglePlayServicesAvailable() {
        int status = GooglePlayServicesUtil.isGooglePlayServicesAvailable(this);
        if (ConnectionResult.SUCCESS == status) {
            return true;
        } else {
            GooglePlayServicesUtil.getErrorDialog(status, this, 0).show();
            return false;
        }
    }

    @Override
    public void onConnected(Bundle bundle) {
        Log.d(TAG, "onConnected - isConnected ...............: " + mGoogleApiClient.isConnected());
        startLocationUpdates();
    }

    protected void startLocationUpdates() {
        PendingResult<Status> pendingResult = LocationServices.FusedLocationApi.requestLocationUpdates(
                mGoogleApiClient, mLocationRequest, this);
        Log.d(TAG, "Location update started ..............: ");
    }

    @Override
    public void onConnectionSuspended(int i) {

    }

    @Override
    public void onConnectionFailed(ConnectionResult connectionResult) {
        Log.d(TAG, "Connection failed: " + connectionResult.toString());
    }

    @Override
    public void onLocationChanged(Location location) {
        Log.d(TAG, "Firing onLocationChanged..............................................");
        mCurrentLocation = location;
        mLastUpdateTime = DateFormat.getTimeInstance().format(new Date());
        updateUI();
    }

    private void updateUI() {
        Log.d(TAG, "UI update initiated .............");
        if (null != mCurrentLocation) {
            String lat = String.valueOf(mCurrentLocation.getLatitude());
            String lng = String.valueOf(mCurrentLocation.getLongitude());
            tvLocation.setText("At Time: " + mLastUpdateTime + "\n" +
                    "Latitude: " + lat + "\n" +
                    "Longitude: " + lng + "\n" +
                    "Accuracy: " + mCurrentLocation.getAccuracy() + "\n" +
                    "Provider: " + mCurrentLocation.getProvider());
        } else {
            Log.d(TAG, "location is null ...............");
        }
    }

    @Override
    protected void onPause() {
        super.onPause();
        stopLocationUpdates();
    }

    protected void stopLocationUpdates() {
        LocationServices.FusedLocationApi.removeLocationUpdates(
                mGoogleApiClient, this);
        Log.d(TAG, "Location update stopped .......................");
    }

    @Override
    public void onResume() {
        super.onResume();
        if (mGoogleApiClient.isConnected()) {
            startLocationUpdates();
            Log.d(TAG, "Location update resumed .....................");
        }
    }
}

Google play services required: