I've selected the TPA3118D2 to power 2x 30 watt speakers @ 8 ohm. The data sheet specifies that it can accept 4.5- to 26-V, but I don't understand what amperage or dc voltage I need to supply to meet my max requirements.
Do you want all of the 30W power? In that case you need 24V. Ideally a bit more (will allow headroom to reduce distortion at 30W).
P = IV --> Power = Current x Voltage
Power will be a bit more than 2x30 due to efficiency. Let's assume 70W (worse case - this would be very poor efficiency for a class D amp).
I = P/V
Current = 70/24 = 2.9A. So you would probably want a 3A capable power supply at least.
Additionally, I don't understand what dB I should set the amplifier to in order to get the 30w output. I could use the example application in the datasheet, but I have no idea if this is enough gain for the wattage of my speakers.
It all depends on the input amplitude. To get 30W out of a 8 ohm speaker, you need 15V RMS:
P = IV = V^2 / R
V ^2 = PR = 240
V = sqrt(240) = 15.5
Line-level audio is 0.3V RMS, so you need a gain (linear) of 49. In decibels, that is 33.8dB (you can use this online calculator
http://www.sengpielaudio.com/calculator-gainloss.htm).
Looking at page 14 of the datasheet, you can set the gain to be 32dB, which is pretty close.
Finally, how would I go about enabling volume control?
You can probably find volume control schemes online. Some will involve an op-amp buffer. You may want a buffer, depending on the source of your audio.
The simplest volume control is a single potentiometer. Connect ground and audio in to the outside terminals. Audio out to the middle terminal (this goes to the amplifier).