They essentially mean the same thing. The difference is in how the actions of the speaker are viewed. If someone says "I plan to learn something", it can imply that they are in the process of planning OR that they are GOING to plan, and people typically take it to mean the latter. If someone says "I am planning to learn something", their actions are more immediate. The ambiguity of a timeframe has been removed; they are now necessarily in the process of planning to learn something. However, as I said, the meaning is the same. They have a plan, and that plan is to learn something. The semantic difference is so slight that even many natives wouldn't be able to explain it to you, so consider them, for all intents and purposes, to be synonyms.

The simple present is used to refer to habitual or repeating actions, facts, etc. whereas the progressive present is used to refer to actions that are in the process of occuring. Back to your question: Correct: I plan a visit to my grandmother's every year, but almost always, something interferes and messes everything up. Wrong: *I plan a visit to my grandmother's right now. Correct: I am planning a visit to my grandmother's right now. What do you mean by 'plan to learn' though? Do you mean something like 'intend to learn'?

There is almost no difference between the two.