What resistor values would I need to use with an LM317 to step 9v DC down to 1.5V DC? Are the two capacitors in the data sheet diagram absolutely necessary?
According to the datasheet, the LM317 requires as much as 10ma of current to flow from Vout to the ADJ pin for proper operation. R1 is normally about 120Ω to get that.
With a smaller R1 you will need a different R2. Use the formula on your drawing.
If you are using a PP3 "transistor" battery, it will have a short life, and you will spend a lot of money buying new batteries. Most of the power (~83%) will be dissipated in the regulator as heat instead of doing useful work in the load (~17%).
You would be much better off to use different batteries. AA Alkaline batteries would not even require a regulator - unless you really need 1.5v exactly.