OK, this is how you wire the relay module to make use of its isolation:
(Two relays or four, no matter. 
)
Now note - you can use the same 5 V to power the relays as the Arduino, but you need to keep the wiring to each part separate and keep the wiring to each part together as a bundle, the control wires and the "5V" to "VCC" as one bundle from the Arduino and the "GND" and "JD-VCC" powering the relays themselves, as another bundle direct to the output terminals of the 5 V power supply.
And you run a separate pair of 5 V and ground wires (such as "figure 8" cable or ribbon) from the output terminals of the 5 V supply itself to the Arduino.
The 5 V supply with transformer, bridge, capacitors and 7805 with heatsink is a trifle quaint. USB "phone chargers" which provide regulated 5 V DC at an Amp or more are pretty common nowadays, even in second hand/ "Op Shops".