Given the likelihood that your current system will not reveal the difference between cables, you’re much better off focusing that nervous energy and money on a speaker upgrade. However, this is a great time for you to decide whether you want to start down the audiophool path. That “bigger” decision can also be made even if you do decide to upgrade your speakers. For example, there are many brands/models of speakers that will improve greatly the sound of your current gear and any comparable gear you get to replace it (by “comparable,” I mean that you decide not to spend your days worrying about whether your system sounds its best).
I advise you to consider carefully whether you want to get to a place where your system resolves cable differences, because the journey is expensive, makes it more difficult to relax and enjoy your music along the way, and is filled with tons of dumb mistakes to make.
Expense. It is difficult to give you a definite amount of money that you can spend to get to being able to discern cable differences. This is because system synergy is a very real factor in the performance of the gear. I’m sure there are $10k amps that could be paired with $15k speakers and $5k sources that will not work well together enough to resolve the difference between something normally obvious like silver vs. copper speaker cables. Then again, there is probably a $1k system out there that will show that difference readily.
So… One of the mistakes you can make is to upgrade one component with the expectation that you will eventually upgrade others to match the quality of that first upgrade. It’s definitely doable, but it is also one of those places where you can make mistakes. Maybe you like the sound of a particular technology or brand or design when you hear it at a dealer showroom. If that choice is hard to match with other components (low efficiency, Lowther drivers, wild impedance curves, for speaker examples), then you will have a hard time, later on, matching your next upgrade, just because of availability, and that will be further frustrated by tastes that you might develop along the way to that second upgrade point. Also, you might bring home a component that doesn’t work well at all with the rest of your gear, and you’ll have to wait for the next upgrade while listening to something you might not like.
Anxiety. Once you start listening to your gear, as opposed to your music, there is a good chance that the sound of your gear will prevent you from enjoying your music. That’s a terrible betrayal of your music and your happiness. The best way to avoid that is to research your upgrade carefully and listen to the upgrade on your system before buying (nearly impossible these days).
Mistakes. Buying gear with features that you can’t use now but expect to use with later upgrade - it won’t happen, dude - don’t do it. Buying gear that WAY outclasses the rest of your system - you won’t get your money’s worth until you make the next two upgrades, and you might not like what you hear. Spending money and time on tweaks - they’re not worth the cost or effort until your whole system is done with the upgrades (and then maybe not). Buying expensive cables - waste of money until you system can resolve the differences, and then the differences will not be huge - stick with entry-level cables until you can hear the differences in cables loaned to you by your dealer (and you are willing to pay those prices for those small differences). Bad speaker placement - a $500 system, with speakers placed for acoustic compatibility with the room, will outclass a $10k system with bad speaker placement - if you’re (and your family) not willing to allow you to (kinda) take over the room, it makes no sense to spend more on your stereo than you did on your wife’s engagement ring (speaker and listener placement are critical to the performance of all the gear). Falling in love - you will go to a dealer or a show, and you will hear something that kicks your ass, and you will buy it, and it will not work with your system, and you will have to start over. Tubes - they can sound magical, but they are expensive and a pain in the ass - my next upgrade will lose the tubes, because I’m just sick of fiddling with them, replacing them, hearing the degradation of their sound over time, etc. - don’t jump on the tube wagon until you’re 100% sure, because they will limit dramatically what speakers you can use.
If you’re going to enjoy the ride described above, then go for it. A lot of folks do. There’s nothing wrong with enjoying the gear more than the music. It’s your life, and to say that geeking out over audio gear at the expense of musical enjoyment is immoral or unhealthy is distinctly unfriendly. Maybe it makes sense in North Korea to expect to be told what to like and dislike, but the free peoples of the Roon forums (fora?) should not be subjected to such mental leg-humping.
If you don’t think you’ll enjoy that ride, but you still want to hear the destination, consider selecting a budget, or saving up to a certain budget, and buy the system all at once. If you diminish your choices somewhat (or a lot, really) you can get great results by working with a dealer. They’ll be able to assemble the system in their showroom, so you can hear what it will sound like approximately (remember room acoustics!). They might even let you take the gear home for a trial period and help you replace components to work better with your room. The aforementioned Rocky Mountain Audio Fest (coming up soon in October) always has budget systems on display, and they are carefully assembled for system synergy. As I recall, last year’s show had a system for as low as $500? Maybe $1000. They go all the way up to $5k, I think.
If you’re set on doing one upgrade, then I would agree that the best way forward right now for you is a speaker upgrade. If I was in your position, given my tastes in music, my understanding wife, my limited knowledge of what speakers are out there and what they sound like, and a limited budget, I would buy a pair of Klipsch horn speakers (the models with midrange horns), maybe the Cornwalls. They are known to sound good with the kind of amplification you have, they do well with more “specialist” gear, and they can survive a few rounds of upgrades to your other gear before you’ll jones for nicer speakers. Then again, I’m very biased, and Klipsh is what got me started on this despairing path.
Good luck!