The shunt-regulated power supply serves to isolate the last power supply capacitor so its quality becomes less important, but that bypass cap across the gas regulator tube is important at the higher audible frequencies. We used 0.056uF teflon across the 0D3 with good audible results. However, I will caution that we have run across teflon caps that never sounded good, even after more than 500 hours of burn in. You still have to use your ears!
For a stock Foreplay III, the plate and cathode load resistors (22.1K 1 watt) are probably the most important in the circuit. We use metal film where there is significant current because most carbon is noisy; quality carbon film can be better than carbon composition but I still avoid it. The traditional gold standard for this application is non-inductive wirewound; I imagine the Shinkoh tantalums would work pretty well.
Of course the input attenuator is made of resistors and they are significant. We use nice quiet metal films; among the best of these (too expensive for a cheap kit!) has been the Dale RN series which have copper leads. Goldpoint, which makes good relatively inexpensive switched attenuators, say they have had good results with certain surface-mount resistors. I have a set of them but they are not yet installed so I can't yet report what my own ears hear. For an inexpensive test, you could just replace the gain-setting series resistor and see what you hear.