Amp Upgrade - Capacitors, Resistors or both?

I own a old (circa 1980s) Counterpoint SA 12 and love its sound. On various threads here, I learn that to get really good sound from an old amp, internal parts need to be replaced.

First, is this true? I've nothing to compare against, so don't know if I'm missing anything. If I need to, should I think about having the capacitors replaced, or the resistors, or do I need both?

Not sure what this would cost, but  knowing what I need to do would be a starting point. Ideas, anyone?

Yes it is more or less true. Electrolytic capacitors in particular deteriorate and have a safe life expectancy of perhaps 10-15 years (more recent ones from mainstream manufacturers are better). Do not take chances with those in your power supply because if they fail you can blow up your speakers and/or have them leak onto your system board. So have at least the power supply caps replaced - this is easy. Modern electrolitics tend to be more compact, so you may be able to fit larger capacities in the same amplifier case for a bit more dynamic headroom (this can be audible, particularly with smallish amplifiers). Resistors also deteriorate but that is pretty harmless. A good design allows for wide tolerances in components, so sonically there is not too much to worry about (unless you are an audiophile hypochondriac).
I am using various older Quad amplifiers and these have all been refurbished professionally by a former Quad NL service engineer. He had a simple policy of always replacing all resistors and capacitors for a complete refurbishment: the components are mostly dirt cheap and time to test them is money wasted. Avoid boutique parts.
Generally speaking, capacitors should be replaced after a few decades.
Resistors, I can't say.
If you give your location, maybe some members can recommend an amp specialist near you.

Replacing resisters is actually pretty uncommon unless the unit in question has noisy parts. Resisters fail open which is usually a pretty safe mode of failure. Caps are always the most problematic, especially aged tantalum caps, followed by diodes. Glue is another issue in some older gear. Some of it oxidizes and becomes both corrosive and conductive. They love to glue caps on with that stuff. Unless you know the glue to be safe, it should be meticulously removed. 
Most of the guys that do this work are pretty good. Avoid a clown named Noah in Wisconsin.