The most general approach is to get rid of the delays using the techniques demonstrated in the 'blink without delay' example.
For something this simple, there is a quick hack which will achieve what you want. Replace this:
int sensorValue = analogRead(A0);
Serial.println(sensorValue);
delay(60000UL);
With this:
for(int i = 0; i < 60; i++)
{
int sensorValue = analogRead(A0);
Serial.println(sensorValue);
delay(1000UL);
}
(Similarly for the other delay() call.)
Just bear in mind that this approach does not extend at all well, and in general if you want to do more than one thing at once you really need to get rid of the delays.