I've done brief research and found out that usually "learn" is not followed by gerunds but sometimes it is. For example:
I am learning boxing.
Is it true? What other gerunds can be used with TO LEARN?
Learn English – Can the verb LEARN be followed by a gerund
verbs
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Forget the "order of tenses". There is a natural sequence past → present → future. Perfect forms, however, do not designate past events but current states brought about by the past events. A present perfect designates a state which is current 'now', and the events which brought about the state may have happened at any time before that, before or after some other event designated with a simple past.
Would, as you rightly observe, is a past form used with present reference to signify that the action represented by the verb sequence it heads is 'counterfactual' (or irrealis, or impossible). Would like consequently designates its object as something now wished for which now, in the present, has not happened.
The phrase to have seen is not, strictly, a present perfect — it is, rather, a perfect infinitive, the marked infinitive of HAVE + the past participle of SEE. It has no tense, and therefore can be employed in a complement to a verb with any tense.
Accordingly, "The fans would like to have seen some improvement this year" may be paraphrased as
The fans wish that they had seen some improvement this year[, but they didn't see it].
English verbs have a past tense form and a past participle form. With regular verbs, both end in -ed. There are many irregular verbs where the past tense and past participle are different, such as wrote and written.
The past tense form is an actual verb and expresses a past event. It will have a subject and objects or complements like any other verb.
I walked to the park.
The past participle form can work as an adjective to say that the noun has been affected, processed, or transformed by a completed action. It's not a verb when used like this.
The destroyed buildings formed an ominous landscape.
Buildings is the subject and formed is the verb here, and destroyed is an adjective describing buildings.
Past participle forms can be combined with forms of to be (for passive voice) or to have (perfect) or both, in this case it's best to think of them as a single verb that takes two words to express.
The buildings have been destroyed.
Have been destroyed is the verb and buildings is the subject.
Much of the time {past participle} + {noun} = {noun} that is {past participle}.
The written evidence - The evidence that is written.
The destroyed buildings - The buildings that are destroyed.
Best Answer
For example: I am learning painting or cooking.
But we generally say: I am learning to paint or play piano.
Any gerund denoting something you learn can be used. However, it sounds somewhat immature or childish (to my ear).
When presented with: I am learning to cook. versus I am learning cooking., the first sounds more mature.
Any activity, basically, that you learn can be described with a gerund and come after learn.
I'm learning typing. Yep, I am.
But what would I actually say in my every day usage?
Call if what you like if you don't like my distinction. It is definitely less elegant.