Wire size and Voltage-Amp relation
If wire size correlates with Amp, and if Voltage correlates with Amp reversely (?), ...


Since \$ P = VI \$ (volts x amps) then for a given power your statement is correct.


  ... why do we have electric lines (for transmission) with high voltage, but also wires with big radius?


Because we are transmitting very high power. Both V and I are high.


  I remember reading somewhere transformers "carry" power, 


Either your memory is faulty or what you read is incorrect. Transformers don't "carry" power. They transform voltage and current from one level to another.


  ... so the power is constant 


No, power depends on the load. For a given load the power in = power out + losses.


  Don't conductors still carry power even if it is lost?


It is not clear what you are asking here.