What type of cables make the most difference

Well shielded interconnects are the most important if your system has multiple components.  XLR renders shielding a done deal, but unbalanced cables vary widely in their noise rejection.  Next in line are speaker cables, which need to introduce low enough series resistance to preserve amplifier damping effect.  Power cables are of dubious value as an upgrade, IME.  Granted, my system is not mega bucks and my room is not perfect and my ears are not young anymore.  My cables are a mix of things I've accumulated over the years from Nordost Purple Flare speaker cables to WBC XLRs.  I do have an AQ Coffee digital coaxial and several Mackenzie unbalanced ICs.

Stick with the power cable provided by the manufacturer and spend your money elsewhere. If you have dirty power spend, your money fixing that. The only way a power cable will improve a components performance is if you upgrade your power lines from your wall outlet all the way to the electrical transformer in your neighborhood, and even then, that wiring will just be a huge antenna that will pick up all kinds of noise. 

I think the most "correct" answer depends heavily on your cabling starting point(s).

If you started at THE most basic level (the interconnect junk that comes with many components, a basic roll of 16g speaker wire, and a Home Depot multi-outlet power cord) then I would rank audibly noticeable 'improvement' as follows:

 1st place: Speaker cabling 

2nd place: Interconnect cabling

3rd place: Power cabling

You are also going to get the most noticeable LEVEL of improvement switching out from these starter cables to even the most basic level shielded / higher gauge speaker cable / interconnects / almost any 'conditioning' power cabling...

As you go up the ladder in quality (and cost), the Law of Diminishing Returns will come into play; subsequent levels of 'improvement' (or even sonically noticeable differences) will become ever more subtle. Then of course, you have to factor in the individuals personal taste and, many times, the physical limitations of the individuals hearing and ability to discern minor differences in sound quality. I find more often than you might think an individual (especially those of us past a certain age) wastes time and money (IMO) chasing specs that they themselves can no longer hear due to age - related hearing spectrum loss...

One area that often is overlooked is asking the question "are my components currently fully capable of performing at their rated capacity / SQ..?" Are the internals clean and dust free? Do all potentiometers work without issue? Do my speakers / amplifiers / media players need recapping or other restorative work to truly reach their stated specifications? As I personally favor 'vintage' equipment, this question is a must-ask in my niche group; however, even for those with newer era equipment, the area of component health should still be explored (but often is not). 

IMO, in the end it really becomes (at its simplest form) a matter of how far you want to chase perfection - and how much time and $$$ you are willing to invest in the chase...

I will agree speaker cables have the most substantial impact on system performance especially when cables are mixed with different brands.

For me and my system, I'd say that speaker cables and interconnects are both important.  It would be hard to rank them.  I never really expected big differences in speaker cables until I got the Silversmith Audio Fidelium cables.

I don't have the ultimate in power cords, but I think they are pretty good.  I've never noticed big differences when installing them.  Perhaps I have clean power here and they have less impact for me?  IDK.