What’s the best way to educate myself (and determine which hardware to use) on how to set up LED light strips that are controlled only through Wi-Fi by home assistant?
WLED.
I am assuming you mean RGB strips like the WS2811.
You can buy controllers with WLED already installed and you can control it from Home Assistant, or you can install WLED firmware on any ESP development board. DIY would cost $2 to $5 each while OTC would run more.
If you mean a single-color light like under-counter lighting for example, again there are several WiFi switches that you can control through Home Assistant. Again, DIY would start around $5 and OTC would go as high as $20.
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WLED is what you’re after, I think.
The @system suggested a very nice piece of hardware back in 2022 which is still widely available for less than $5 when directly purchased in/from the country of manufacturing 
As such a device is completely own-able the owner(!) is free to decide what *ware to run (for example kno.wled.ge or esphome.io)
And for a knowledge resource as well as top notch controller hardware @Quindor got you covered 
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What would you suggest for something very quick and easy for a few more $ (plug and play)? Are these compatible with HA wled integration?
I ended up going with this. Maybe not perfect but good enough for my use case. I would likke to do more with WLEDs however in the future 