What is the minimum radio and navigation equipment for required for VFR and IFR certified aircraft?
In the US, this in in 14 CFR 91.205, Powered civil aircraft with standard category U.S. airworthiness certificates: Instrument and equipment requirements. 

There's a lot of detail so you can check the whole thing, but looking only at communication and navigation items, the required equipment is:

VFR (including night VFR)


Magentic direction indicator
Altimeter (for airspace navigation)


IFR


Instruments and equipment as for VFR
Two-way radio communication and navigation equipment suitable for the route to be flown
Rate of turn indicator (for timed turns)
Directional gyro
A clock (this question explains why a clock is needed for IFR navigation)


That list might seem very short, but remember that under VFR there's no requirement to have radios and there are even some vintage aircraft flying around with no electrical system. 

Under IFR it's fairly obvious that you need a radio for ATC communication and whatever navigation equipment is needed for your planned flight, e.g. you can't fly a GPS approach unless you have an IFR-certified GPS. But there are multiple approach types and they all have different equipment requirements so the regulations just give a general statement.

Note that commercial flights operating under parts 135 or 121 will have additional requirements; the information above is really the bare minimum that's legally possible for GA flights.