覚える: to put something new into your memory. Like "Learn 10 kanji a day."
思い出す: to retrieve something which you learned before. "I remember the last time I saw him - in 1985 on a beach."
記憶: one's memories
習う: to learn, to take lessons in (ie the guitar) (not sure if it can be academic as well)
学ぶ: to study in depth (ie your major in college, a langauge)