wow, great subject line !
first off, relays have 3 separate power supplies.
you already know that the contacts is completely isolated from the controls, so that one is moot.
you have a power supply for the coils. relay coils can be any voltage, AC or DC, all depends on how they coil was wrapped and the materials used on the contacts.
the third is where your question lies. how to send a signal to activate the circuit that powers the coils.
there are many links on how to control a relay and using some form of FET or transistor is how you use a small signal to control a large one.
if you are buying any of the relay boards that have the opto-isolators built in, then your power requirements are no more than 20mA max, often much less.
so, you need a clean power supply for your logic stuff.
either some form of filter, isolation or separate power supply for your low voltage power control
and whatever you have or need for the contact side of the relays.
using a cap at the input of the arduino goes a long way to filter the voltage by relays being switched. a hefty switching power supply will have large caps to allow for the surge in the relays switching.
Start out with the simple connections, make sure you tie your DC ground together, and all should be good. only if you have problems should you need to make any special adjustments.