You can’t “learn Arabic” without learning how to say “to learn,” amirite?
Sorry, let’s just move on to the vocabulary.
We are going with “learn” here in the sense of acquiring new knowledge or skills by study, not “learn” in the sense of “found out,” like “I just learned that your cat had kittens!” They are different concepts and translate differently.
“Learn” in its purest “acquisition of knowledge” form translates as the fifth form verb تَعَلَّمَ (taʿallama). The form I root is عَلَمَ (ʿalama), which means “to know” and makes all kinds of great words that we can talk about on other occasions. Recall that to derive a form V verb, you take form II of the root (here عَلَّمَ, ʿallama), and add a “ta” prefix. Form II verbs mean “to cause someone to do (form I),” so in this case “to cause someone to know” or “to teach,” and form V verbs are the reflexive form of that, which here would be “to cause oneself to know” or “to learn.”
“I am learning Arabic” = أتَعَلِّمُ اللُغةَ العَرَبية (ataʿallimu al-lughah al-ʿarabīyah)
“Yesterday they learned the names of the planets” = أمس تَعَلَّموا اسماءَ الكَواكِبِ (ams taʿallamū asmāʾ al-kawākib)
“Education” = تَعَلُّم (taʿallum), the verbal noun of تَعَلَّمَ, can mean “education,” although the more common words are تَعليم (taʿlīm) or تَدريس (tadrīs), both verbal nouns based on the two form II verbs that mean “to teach” (see next entry).
“Student” is totally unrelated: طالِب (ṭālib, fem. طالِبة ṭālibah, “asker” or “demander”; the student “asks” knowledge of the teacher). If your mind immediately went to “the Taliban,” that’s because “the Taliban” began as a student protest movement so the word ṭālibān means “students.”
NOTE: Sometimes you will find the form I verb دَرَسَ (darasa) used to mean “learn,” but this is an imprecise meaning of this verb, which really means “study,” not “learn.” I guess you could argue that it means “learn” in the sense of “the process of learning,” but I think if you want to be clear and accurate, تَعَلَّمَ is the better choice (although I make no claims about speaking; colloquial usage trumps everything).