First off, welcome to the forum. For a quick guide to buying capacitors in general -
1uF (microfard) = 1,000nF (nanofarad) = 1,000,000pF (picofarad)
To read the cap codes, e.g. "472", take the first two digits (47-) and append the number of zeroes indicated by the last digit (so 4700): This is your cap value in picofarads, so "472" = 4700pF = 4.7nF = 4n7. 221 would be 220pF.
As the capacitance of the cap gets bigger, so does the physical size, although there are some caveats.
For 2uF and above, or where specified, use polarised electrolytics. They have a big capacitance for a relatively small size. Most places sell the "radial" version (i.e. both positive and negative leads emerge from the same end of the can) and most layouts will have the correct spacing for this type. Note that you have orient these caps with the negative led (indicated with a stripe on the can) the correct way around: Most layouts will also indicate this. You want to make sure the voltage rating is at least
16v for a 9v circuit. Anything bigger is fine.
For the nanofarad values, "box" caps are usual, whether polyester film, polyester mylar etc etc. They don't tend to come in much choice as far as voltage ratings is concerned, but are usually rated way above 9v so it's not a concern
For picofarad caps, ceramic are the usual choice. Small, cheap, and again, the voltage rating is of little concern.
If you look at the picture of someone else's chunky cheese pcb below, you can make out the blue cylinder electrolytics, the box caps for the nanofarad stuff, and the two orange ceramics for the pico values:
Check my notes
for the parts list you posted.
C1 - 47nF - 473
C2 - 1.5 - 152 (this is 1.5nF)
C3 - 47nF - 473
C4 - 47nF - 473
C5 - 47PF - 470 OR 47
C6 - 4.7uF polarized electrolytic
C7 - 47nF - 473
C8 - 6.8 nF - 682
C9 - 2.2nF - 222
C10 - 10 uF - 103 this is actually 10nF
C11 - 47nF - 473
C12 - 4.7uF polarized electrolytic
C13 - 22uF polarized electrolytic
C14 - 22uF polarized electrolytic
C15 - 100pF - 101
C16 - 100 nF - 104