I learn the taxa around me in a continuous cycle: I see an organism, I take a picture, I ID it either through a rough-ish ID from iNat CV and then refine it by looking at ID guides, or if the CV fails me, then I key to family, genus etc. until it become impossible with the level of detail I have. When I see the organism again, I tend to remember it at least to genus or by its colloquial name. And over time, I get to know the common species in my area.
However, the occasional taxon that I particularly like, like birds, I tend to learn their higher taxa on a broader, more global scale, but not the specific species. For example, I may know that that is a new world warbler, but not which type of new world warbler, or I may know that’s a tern, but not which species of tern. (The myriad of birds of the world encyclopedias really helped with that ;)). Occasionally, if its a particularly distinct species, I tend to know it to species.
Same goes for most insect groups but to a lesser degree, so I know them to the UK or European level, like I may know that it is a type of blowfly, but I wouldn’t know which species of blowfly.
It really depends on the simplicity of the taxa too. Like birds are more well documented, so there are more field guides and whatnot, so most of the time, species ID is very easy, but with insects there are many groups that cannot be easily distinguished, or species that can only be told apart by minute details.
So overall, its mostly about preference. If you prefer one family over another, you may decide to learn the genuses in that family, for other taxa you may want to learn 1 or 2 genuses and leave the rest because the details are so limited for ID. :P Who knows, you may also want to look into the few handfuls of research in the ID of some obscure genuses and become the leading IDer of them genuses.