You can either connect the 7809 to the 7812 output or the 7812 input. There are advantages and disadvantages with each approach.
Connecting it to the input:
ripple will be higher since it sees the power supply ripple across the 1000uF caps
the power dissipation in the 7809 will be the unregulated input voltage minus the 9V output voltage times the output current, plus about 5mA times the unregulated input voltage. The 7812 will be unaffected.
Connecting it to the output:
ripple will be lower since it sees a regulated input
the power dissipation in the 7809 will be 3 volts times the output current plus about 60mW (5mA times 12V)
the power dissipation in the 7812 will be higher by the unregulated input voltage minus 12V, multiplied by the 9V output current plus about 5mA. Thus the total dissipation in the 7812 will be (Vunreg - 12V) * (Iout12 +Iout9 + 5mA) + (Vunreg * 5mA)
To put numbers on this, if you're drawing 100mA from the 12V supply and 50mA from the 9V supply, and the unregulated voltage is 17V then you'd have (if I did the arithmetic right, E&OE):
In the first case you can get away without heatsinks on either regulator in most situations, in the second case the 7812 should probably have a small heatsink (especially when you consider the possibility of high line voltage).
In general, if you're connecting a regulator to another regulator you also have to worry about the dropout voltage. In this case, there is 3V difference available which is fine, even for a 7809, but if you tried to run a 7805 from the output of a 7806 there would not be enough voltage for the 7805 to operate correctly.