Which power cord to upgrade first?

I don't think there is a "right" answer. In my case I find it to be on the DAC and preamp. My advice would be to buy used and experiment.

Amplifier OR amplifier AND power conditioner (if you're using a power conditioner.) These two cords should be no less than 10AWG wire size. A floating shield (connected at the AC plug end only) is a nice option, but most modern equipment has internal RFI/EMI filtering.

Front end stuff can use smaller gauge PCs, but no less than 14 AWG.
Exception: DACs seem to benefit from larger conductor sizes like 10 AWG. They're not power-hungry devices, so I don't know why this is, it just is.

Any device that has digital circuitry of any kind (DACS, CDT, CDP, HT processors, etc) should definitely have a shielded power cord, and preferably use a circuit that is separate from other (non-digital) front end stuff (tuner, tape, preamp, phono preamp, etc.)

I've never heard a bigger difference than when changing the PC on the source, much more so than the amps in my experience. Obviously, others have heard greater shifts elsewhere. As Wellfed suggests, buy used and experiment.

In my system,the source (CD player) cord upgrade had the greatest improvement.
The power amp and preamp cord improvements were subtle, the CD players improved cord was substantial. From what I read, the CD player can 'broadcast' digital noise through the power cord corrupting the other components. The improved cord keeps the digital noise from polluting the rest of the chain, along with the the improved power delivery.

CD player = Musical Fidelity A5 with JPS labs digital AC
Preamp = Adcom GTP750 with VHAudio flavor 2.
Amp = Conrad Johnson MF2250 with VHAudio flavor 3.

It might depend on the relative quality of your components. For
example, if you own a Rotel CD player, and an Ayre amplifier, you might
hear greater benefits when changing the amplifier's power cord and vice
versa due to the better power supply and topology of the higher level
component.

As synergy is the key, and as there is no right answer, experimentaion is
important, and if you have one higher grade component than the others,
I'd suggest starting the experiment there.

What components do you have?