Question: > 
> *I want to go home.*
> 
> 
> 

Here the word *to* belongs to what part of speech?

Answer: 
It depends on who you ask. Within the confines of traditional *parts of speech*, the present-day *to* in the infinitive may be a preposition or it may a particle.

Even in the 19th century, grammarians were wary of assigning the to-infinitive to a single part of speech. For example, if you look up *preposition* in George Pliny Brown's 1899 book *[Advanced Pages of Elements of English Grammar](https://www.google.com/books/edition/Advanced_Pages_of_Elements_of_English_Gr/jRFMAAAAMAAJ?hl=en&gbpv=1&pg=PA59&printsec=frontcover)*, you would find nothing resembling the to-infinitive. Instead, "The Infinitive" would be listed under the hedging header "Words Sometimes Called Parts of Speech" with "The Participle," "The Article," and "The Gerund." Brown describes the infinitive as having some functions of both a verb (in how it can be modified) and a noun phrase (in how it can be the subject of a sentence). In other words, Brown is illustrating the traditional grammarian's difficulty to reconcile the structured *parts of speech* with forms that don't fit well in the existing theory. The *to*-infinitive, let alone *to* itself, is right at that fault line.

The Oxford English Dictionary reflects the difficulty of defining *to* before an infinitive under ["to, prep., conj., and adv.," definition B](https://www.oed.com/view/Entry/202695). The full explanation is too long to quote in full, but I want to highlight two elements of that explanation:

1. The *to* preceding the infinitive comes from Old English, where it would have been considered a preposition (*tó* [[see Bossworth Toller]](https://bosworthtoller.com/30587)). An example of the preposition-based usage with a verb of motion (*come to*) is under I.1.a.(a) from Bede: "Monige cwomon to bicgenne þa ðing" [Many came to buy the thing]). In that kind of usage, even today, the OED would consider *to* a preposition: "But after an intransitive verb, or the passive voice, *to* is still the preposition."
2. In other uses of the *to*-infinitive not evident in Old English, e.g., using the infinitive as a subject, *to* is considered to have "lost all its meaning, and become a mere 'sign' or prefix of the infinitive." Though the OED wants to call this *to* a prefix, a morphological attachment to the word, that description is quite close to the definition of a *particle*, a word that doesn't fit conveniently into other parts of speech or classes of words ([Wikipedia](https://en.wikipedia.org/wiki/Grammatical_particle#In_English)).

It's worth noting that the OED is not an authority on grammar, and its primary purpose isn't creating rigorous classifications of language use. Nonetheless, it represents what later systems will generally show: while the infinitive *to* once functioned as a preposition, in modern English that language is no longer sufficient to describe what *to* is doing in many infinitive uses. Hence it is often called a *particle*.
'To' in 'to kick' is, according to Wikipedia, an infinitival particle. The infinitive 'to kick' without 'to' so just 'kick' is called the bare infinitive, whilst 'to kick' is called the full infinitive.