¡Hola a todos!
This is a great question! Learning pronouns in another language can be a significant challenge.
This is a short breakdown of the different parts of speech you mentioned:
Tú
Tú is a subject pronoun, which means that it serves as the subject of a sentence. So, as you pointed out, the subject pronoun refers to the person or thing that is carrying out the action or verb of the sentence. For example, in the sentence Tú haces la cena. or “you make dinner,” tú or “you” is the subject pronoun because it is “you” who is performing the action. However, unlike English, Spanish is a pronoun-dropping language. This means that the verb, in this case, haces, “you make,” is conjugated in the you singular form of the present tense, and as such, the verb indicates “you” without the subject pronoun. In English, dropping the pronoun completely changes the meaning of the sentence; however, this is not the case in Spanish.
It is also worth noting that tú is the informal “you”, and usted is the formal “you” singular. There is some more information on formal vs. informal in relation to verbs in lesson 2.8 Action in the Present Tense.
Tu
Tu written without an accent on the u is not the same as tú with an accent. As you correctly pointed out tu without an accent is a possessive adjective, meaning it is translated into English as “your.” For example, Soy tu padre. “I am your father.” Tu is singular and informal.
Tuyo
Tuyo, like tu, is a possessive adjective; however, it is a longer-form possessive adjective than tu. Long-form possessive adjectives must go after the noun, not in front of it. For example, in the sentence Es la hermana tuya. “(She) is your sister.” tuya “your” must follow and agree with the noun hermana “sister.” There is more information on this topic in lesson 4.8 Talking About Yours and Mine.
Te
Te is an object pronoun. It can be used as both a direct object pronoun and an indirect object pronoun. A direct object pronoun can be used to replace the object in a sentence that receives the action of the verb. For example, in the sentence Estoy cocinando la cena. “(I) am cooking the dinner” it is the dinner that is being cooked; therefore, “dinner” can be replaced with a direct object pronoun to mean “(I) am cooking it.” La estoy cocinando. Te is a direct object pronoun, and therefore, the sentence “(She) is looking for you.” is translated as Está buscándote.
Te is also an indirect object pronoun. An indirect object is the person (or thing) TO or FOR whom an action is done. For example, in the sentence Elena te escribe un mensaje. “Elena writes you a message” the “message” is the direct object as it is being written, and “you” is the indirect object because the “message” is for “you.”
Ti
Ti is a prepositional pronoun. As their name suggests, prepositional pronouns must follow a preposition. Prepositions are small words that indicate time, place, location or possession. To, at, in, and of are all examples of prepositions. Therefore, when you have the pronoun “you” after a preposition in Spanish, you should use the prepositional pronoun ti “you.” For example, in the sentence Tiene un regalo para ti. “(He) has a present for you.” the pronoun comes after para, which in this case is translated as “for;” as such ti, “you” is used. For more on prepositional pronouns, visit lesson 4.9 “Between You and Me”: Prepositional Pronouns.
I hope this is helpful! These different language features can be tricky to begin with, but as you progress, you will start to get a good idea of what belongs where.
¡Saludos!
Indy