I use this. It works great after 3 years still. Just make sure to take out any mr. tuffy tire liners or else the friction and heat will give you spontaneous flats.
As far as I know it came with its own quick release rear axle, in fact it had to because it was steel and there's more forces and stuff. That's relatively easy to swap out honestly.
Looking at your bike you have a fixed gear bike, so you might not be able to get the most out of a trainer, but it's something. For me changing gears on my bike also changes the resistance for me on a fluid trainer. It also appears that you don't have a quick release on your bike, so I'm not sure how you might be able to put it onto a trainer, but I think you should be able to. Also, I'd recommend clipless pedals because the upstroke while pedaling really helps you go faster and train other muscles too, BUT IT'S NOT NECESSARY, if you don't care about that stuff or you're trying to save money. If you're aiming to just move your feet then you're fine.
Thank you for the advice. I think I'd want something cheaper than that right now. Are the ones I'm seeing for under $100 probably not able to work with my bike? I also have never seen one of those quick release rear axles before, lol. As for being a fixed gear - I'm pretty sure most of my friends always rode fixed gears, and they're the ones who I first ever heard talking about trainers. So there has to be a way, I guess? I can understand, though, how a fixed gear on a trainer would be less of a workout - since the workout from riding a fixed gear is mostly from having to manually pedal up hills/slow yourself riding downhill.