In most uses, "sich merken" is a good translation for "to memorise". Sich einprägen is less common and means the same. "Ich habe mir das Kennzeichen gemerkt/eingeprägt und werde den Fahrer anzeigen". For sitting down and going through a list of vocables, you would normally say "lernen". It sounds completely casual, it's just the thing you do as a language learner. Whereas sich merken or sich einprägen makes it sound like it's of special importance. -> "Ich lerne diese Vokabelliste". You can throw in "gerade" if you want to stress you are doing this right now. "Speichern" or "abspeichern" means to save, in an IT sense. You occasionally hear it used metaphorically for memorising something. "Dieser Stoff muss sitzen, den sollten sie wirklich abgespeichert haben" - a typical professor's phrase.

Often there's no one-to-one correspondence between expressions in one language and another. But once you've seen enough examples of actual use, you'll get a good idea of the nuances. "Sich merken" is closer to "to remember". "Sich einprägen" means something like "to remember strongly". "Prägen" means "to stamp, to emboss", so the image is "to stamp something into your memory, like one would stamp a picture on a coin". For "to memorize a list of words", I'd use "eine Liste von Worten/Vokabeln (auswendig) lernen", i.e. "to learn (by heart)".