Okay Bro, you are almost there.
<<<I missed your reply and went ahead and built the thing. I left the 220R as it was and changed the R1 and R2 to 2k. I got the numbers using the calculator.>>>
That's cool. So you have a total of 4k out of R1 and R2.
<<<<I paralleled the secondaries of the transformer and it only measured 18.3VAC. I went by the instructions for my transformer, listed here. The part number I'm using is 62063. I don't know what's wrong here. I tried two transformers and they both measure the same. Probably my stupidity, but I don't know where.>>>>
Well, that is what I expect as your transformer is rated at 2 x 17.8v. So your hookup and measurements are fine. The problem is that you need to connect the secondaries in series. Keep in mind that paralleling the secondaries keeps the voltage the same and doubles the available current. Connecting the secondaries in series doubles the output voltage but keeps the available current the same as one side.
<<<<<Speaking of stupidly, I did check the DC volts coming out before I checked the AC volts coming in and it's not that far off on the +24V. It came in at 22.9V. The +48V came in at 24.5 Volts, but I didn't put one of the 1N4007s in, thinking that I might blow the regulator if it was too high.>>>
I would put the 1N4007 back into the circuit, as looking at the circuit it appears to be necessary. After some checking it turns out the LM317 can easily handle the voltage that you are throwing at it as long as it is getting clean DC. Clean voltage will not blow the LM317 out, but overpowering it will. IOW, your transformer is rated at less than an amp and should be okay with the regulator.
Like I said before, I'm not an expert at these matters so anyone else's input is appreciated. But the above is how it appears to me.
Edit: Hmmm, I see I have posted long after others have already replied. That shows about how together I am tonight. My girlfriend is telling me to log off and give her attention, so I gotta cruise for now. I'll look at this thread again tomorrow after work.