Can I add two LEDs to this circuit?
Yes, you can add them in series or parallel.
Series would perform best, as there is a situtation that occurs with this circuit where if the resistance is too low, the LEDs all start to glow dimly all the time.

For a 9V battery application the math to choose the appropriate resistor is as follows:

\$R=\frac{V}{I}\$

Now the voltage (\$V\$) is the battery voltage minus the forward voltage drop of each LED in the string (From Vcc to pin3, or pin3 to GND). Red LEDs can range from 1.2V to 2.4V depending on the manufacturer, power rating and such.
I'll use \$Vf=2.0V\$ and \$I=5mA\$ for this example.

\$R=\frac{9-2Vf}{0.005} = \frac{9-4}{.005} = 1000Ω\$

For your specific LEDs, you'll need to substitute the correct \$Vf\$ for the \$I\$ you plan to use, from the datasheet.

Also, when trying to do this with white LEDs, I had to use a NPN and PNP transistor to drive the top and bottom LEDs, as the resistors were too small to get the brightness I needed and the whole LED string would start to glow all the time otherwise.

This is a pretty safe and simple circuit.  Play with it and try different resistors and different LED combinations.  Doing so can lead to an intuition about how the circuit works.