Depends what you define as 'turned on'. If you plug the computer into a live wall socket and switch the PSU on - the PC is drawing current. Older PC may draw as much as 30/40 watts to keep the motherboard live. Some even draw current with the PSU switch off.
Also turning the PSU switch on causes an inrush of current as the capacitors + other circuitry initially charges. The ATX 2.2 specification allows 100A inrush current @ 110V and 50A @ 230V - yes for a brief few milliseconds your pc could be drawing 11,000W and not even be running yet!
Finally pushing the 'front power button' to boot your PC there should be no spike in power draw. Draw will be dependant on CPU/graphics/hard disk usage.
If you want a cheap way to measure it you can buy a plug-in power and energy monitor/meter from ebay/amazon for ~$15/£10. They will show current, wattage, mains voltage, power factor and if you input cost of electricity will show cost. They also record it so you can see cost per day/month/year.
They won't measure anything advanced like inrush current though.