~ておく also has a more general nuance of doing something for future purposes, i.e. performing an action that will have some benefit down the line.
これも覚えてください is fine grammatically, but it kind of sounds like you're speaking in a vacuum. "Please remember this, too (not for any particular reason, but because I'm telling you to right now)." whereas 覚えておいてください has an added nuance of "Please remember this, too (because it'll serve you well later on)."
In English, this isn't a nuance that is necessarily expressed using a specific grammatical structure, but that doesn't mean it's superfluous in Japanese.
Indeed, when learning Japanese (a language completely different from English in almost every possible way), often you'll come upon structures, particles, phrasing, etc. that can't simply be translated 1-to-1 into English. At times like this, it's not so much a matter of understanding "how Japanese people feel" (which makes it sound like these are arcane concepts that only exist in the Japanese mind, something that isn't really the case), so much as grasping the underlying nuance (even if it doesn't have a clear and straightforward English translation) and internalizing how it's expressed in the Japanese language.
I got a bit philosophical there for a second (sorry about that), but I feel like beginning learners (through no fault of their own) tend to overcomplicate Japanese. If you can gradually learn to "de-mystify" the language (and asking questions like this is a part of the process), it'll go a long way to helping you in your studies. Good luck!