Building a small 4-bit adder on a breadboard
It looks as though a 5V power supply is suitable for the parts you have listed. The easiest way to generate this 5V from a 9V battery is using a 5V voltage regulator such as a 7805. A resistor won't be able to produce a reliable 5V source from 9V, because the current drawn by the circuit will not be constant. The 7805 needs a couple of capacitors close to it for stability (see, for example, figure 6 of this datasheet).

I note that you have a mixture of TTL (74LS...) and CMOS (74HC...) parts. Mixing logic families like this is often not guaranteed to work correctly, due to differing output and input voltages thresholds between the logic families. A very quick look at the datasheets for your parts suggests that high output voltage of the TTL part is possibly as low as 2.7V, whereas the CMOS parts may require at least 3.15V on the input to register a high value. Though this can be fixed with level translator chips or pull-up resistors, I would advise just getting some 74HC32 (or 74AHC32) parts to replace your 74LS32 ones.

See this question for a description of how to calculate the appropriate resistor for LEDs. However, I suspect that your LED current will be limited by the maximum output current of your chips. You can get this value from the datasheets for the logic chips. Note that there is usually a per-pin current limit, as well as a total limit for current provided or sunk by all outputs. 

Though in this circuit you might get away without them, it is always advisable to put a 100nF decoupling capacitor across the power supply pins of each logic IC.