Behind cerco a capire, provo di parlare and voglio di andare there is one task being attempted, which is finding the preposition that fits the meaning of the sentence. There is nothing to work out, and nobody says so out loud. The meaning has no vote. Which small word goes between two verbs belongs to the first of them, the way a gender belongs to a noun, and it is learned with the verb or not at all.
When one verb governs another in Italian, what sits between them is di, or a, or nothing. Di goes with cercare, finire, smettere, decidere, dimenticare, sperare and credere. A goes with provare, riuscire, cominciare, continuare, imparare, aiutare and insegnare. Nothing at all goes with volere, potere, dovere, sapere and preferire, which take the bare infinitive: voglio andare, devo partire, so nuotare.
No rule generates those three groups. There is no shade of meaning that di carries and a does not in the groups above, nothing about trying or finishing or starting that points one way, and no test you can run on a verb you have not met. That is precisely why the question keeps being asked, and why the answer is a small set learned as a set rather than a principle to apply.
Cerco di capire.
Provo a chiamarlo.
Voglio andare a casa.
Ho imparato a nuotare.
Ho smesso di fumare.
Cerco di capire and provo a chiamarlo are where the question answers itself. Two verbs doing nearly the same work, and the prepositions are different, with no difference in meaning to account for it. Put the two lines side by side and there is nothing to explain.
English is no help either, and ho smesso di fumare shows why. The English structure after “stopped” is a gerund, after “learned” it is “to”, and after “want” it is “to” again, so the shape of the English sentence and the shape of the Italian one do not line up even when both languages have something in the slot. A few of the a verbs also take a person before the infinitive, which is worth having as a whole phrase: mi ha aiutato a capire, le ho insegnato a nuotare.
The edge that trips people up
Some verbs take either preposition, and the two readings are not the same, so here the choice does carry meaning, and it is the only place in this post where it does. Pensare is the one you will meet first, and the split there (penso a te against penso di partire domani) is where the post on ci and ne turns, since ci penso and che ne pensi both come out of it. That is the same split the post on penso che sia uses when it reaches penso di avere ragione, where di with an infinitive is the opinion.
Penso a te. Penso di partire domani.
Finire does the same trick with a different pair. Finire di is to finish doing something, and finire per is to end up doing it, which is a statement about where you landed rather than about the activity stopping.
Ho finito di lavorare, e ho finito per restare a casa.
The lists further up are a working set, not the language. Italian has many more verbs that govern an infinitive, and each of them comes with its preposition attached, so the set grows as you meet them. Nothing here closes it, and a page that claimed to close it would be wrong in a way you would only discover later, in the middle of a sentence.
How to hear it
A preposition learned apart from its verb is a decision you then have to make every time, with nothing to make it with. Stored the other way, as part of the verb, it is not a decision at all. Cercare di and provare a are single items, the way a noun and its article are single items, and cercare on its own is an incomplete thing to have learned.
That is also what to do with the next one you meet. When you hear riuscire a finire or decidere di restare, the phrase to keep is the two words together, not the verb with a note about a preposition somewhere nearby. Kept that way, the pair stops being a fork in the sentence and becomes one sound. And for a verb whose preposition you do not have yet, the bare infinitive is not a safe default, because the verbs that take it are their own small group, and a verb you have not met will more often want di or a: the sentence is better rebuilt around a verb you do have than guessed at.
A verb you meet next month arrives with its preposition attached, so the work does not grow as you learn more. What changes is only that more of your verbs come already decided.
