The LM317 is an nice beginner device to experiment with, and can be used for either voltage or current regulation, depending on the design.
Just D/L'd new Fritzing, and took a quick look at your schematic...
R1 & R2 are set-up as a voltage divider. Because they are both the same value, the voltage in between them (at ADJ) is 1/2 of the the output voltage.
The key to working with this device is to realize that it wants to see 1.25V at the ADJ pin, and will vary the output voltage to achieve that. Because your voltage divider is set-up to produce 1/2 of the output voltage to ADJ, then this circuit will try to produce 2*1.25 = 2.5V output.
Notice I said TRY - the problem with your circuit is the LM317 is not a very efficient device, and requires a certain minimum amount of voltage difference between the input voltage and the maximum output voltage it can produce in order to function correctly - this is called the "Drop-Out Voltage".
This minimum voltage difference between the input & output varies by temp. & current, but for the LM317 typically you want at least a 2.25-2.5V difference to be safe.
Your schematic shows only 3V in, and you are trying to get 2.5V out - this will not work right - you need more input voltage.
But you don't want the input voltage to be too much higher than needed, either. This is because the LM317 is what is called a "linear regulator" - the excess input voltage just gets converted into waste heat!
5V input would be just about perfect for your circuit - it is also a very common voltage to find. It is one of the standard voltages a PC power supply produces, it is also the voltage many USB-based power devices produce.
6V battery would also work quite well - you just need to use four 1.5V batteries in series instead of two! ;)
Another solution to this problem - they also make fancier "low drop-out" regulators (called this because they require a smaller drop-out voltage difference), such as the LM1117T-ADJ, these newer devices let you get away with a smaller minimum voltage difference than the older LM317 device.