I've posted a simplified schematic for my power supply. My only concerns are the number of 10000uF electrolytic capacitors I need to use and how I connect them. Is one enough to filter the input voltage of all the rails, including the -15V rail? If not, do I need one for every rail or one for the positive rails to share and one for the -15V rail?

 

I think that schematic is off. If you want plus and minus, you need something more like this.

Submission: Power Supply Circuits

There is something about your configuration (specifically the ground) I'm having problems with. I'm going to have to work it out, but I suspect it won't work as you have drawn it.

The article link basically answers your questions. I haven't finished the article, not even close, but current plays a major factor in what capacitances you use. So how much current do you need total (on both sides)?

 

OK, I didn't bother to trace all the paths, but you are throwing about 3/4 of the power away. The red traces are double the voltage of the orange, which means they will overwhelm the orange. Plus on those paths where the orange is split between two resistors it is half again that voltage. It won't work.

 

The 24V regulator needs an input that is minimum 26VDC. Add 2V for ripple then the recifiers produce 28V. the 5V regulator must dissipate 23W if its load is 1A then it will melt even if it has a huge heatsink.

The peak voltage from the transformer must be 30V then its AC voltage must be 21V on each side of the grounded center-tap. So you need a transformer that is 42V with a center-tap. It might be listed as 21V-0V-21V.