Electrical – What relay for high power circuit, low power controller

Obvious problems: Well yes. As drawn right now, there is some confusion but:

1: Either you don't get 220V to your secondary relay (The one connected to the Arduino) because the line leaving your primary relay going left is a trigger signal line, rather than a power line OR

2: You DO get 220V to your secondary relay and you cannot trigger your first relay because the line leaving your primary relay going left is a power line, rather than a trigger signal line.

My first recommendation would be to redraw your diagram with clear labels on all wires.

The USB<->TTL_serial adapter is being powered from USB.
One of its main jobs is to drive the TX signal line to whatever state is appropriate for whatever communication is taking place.
If there's no comms happening, the default state for a TTL UART TX line is high (so 3.3V or 5V or whatever the device's supply is).
As a result, you're feeding this 'high' into your other circuit (ATmega, adruino, whatever).
So now what does this other circuit do with this 'high'?
Well most microcontrollery things (and many others too) will have some sort of diode arrangement between their I/O pins and their supply lines - either deliberately as ESD protection, or simply as an unavoidable result of the manufacturing process - something like this:

^{simulate this circuit – Schematic created using CircuitLab}

So as you can see, if your USB<->TTL_Serial adapter is trying to do its job and apply a 'high' to the micro's I/O pin, due to the micro's internal diodes it will end up trying to supply power to the whole circuit. Some adapters may be more or less tolerant of this state of affairs than others...