This is a complicated question to answer. The siddur is made up of different prayers brought together from different eras. I will not give you an exhaustive list as that could be the subject of a whole book, but instead give you some examples so you can recognize the literary layers.
The oldest strata of prayers in the siddur are those from the Tanach: most commonly from the book of psalms, but also from other books such as Exodus, Deuteronomy, Numbers, and Chronicles. Some examples of these would be:
- Song at the Sea (Exodus 14:30-15:19)
- Mi khamokha before the Shemoneh Esrei (Exodus 15:11)
- Shema paragraph 1 (Deuteronomy 6:4-9)
- Shema paragraph 2 (Deuteronomy 11:13-21)
- Shema paragraph 3 (Numbers 15:37-41)
- Nehemiah 9:6-11
- Ovadiah 22:29, Zekhariah 14:9
- In P'sukei D'Zimra (I Chronicles 16:8-36, Psalm 100, pastiches of fragments from other psalms, assembled in the rabbinic period)
- Daily Hallel (Psalms 145 - Psalm 150)
- After the psalms of the daily hallel (I Chronicles 29:10-13)
Appended to this most ancient layer are the prayers composed from roughly 516 BCE to CE 120. Sometimes these were new verses created afresh, and sometimes they would be built from biblical verses that were assembled or reworded in new ways. The most prominent addition from this period would be the 19 benedictions that make up the Shemoneh Esrei or Amidah (standing prayer). These are credited to "the men of the Great Assembly" with the 19th addition added by Rabbi Gamaliel II. Also dating from this period are pieces of the two benedictions before the Shemoneh Esrei, "Creator" and "With Great Love", but their text remained fluid through the Gaonic period (roughly 589 CE to 1038 CE), with some parts like the Kedusha of Creator being dated as later Gaonic poetic additions.
The most prominent set of Aramaic prayers is the Kaddish, found in several different forms. Some parts of it may be as early as 100 BCE but we do not have it as a complete written text until the Siddur of Amram Gaon around 900 CE. There is evidence that the contents of it was fluid during this interval.
Another Aramaic prayer on the Sabbath is "Yekum Purkan" which was likely composed in Babylonia near the end of the Gaonic period, but which we do not have a text of before Machzor Vitry from around 1100 CE in France. This is an example of a relatively late addition that is found in the Ashkenazic (Northern European) rite, but not Sephardic (Spanish) or Mizrachi (Eastern) rites.
Much more could be written on this subject. My source for much of this is Jewish Liturgy: A Comprehensive History by Ismar Elbogen, but it is written at a scholarly level probably above what you should be studying right now. The primary takeaway for you is that the siddur was not given as a revelation all at once like the Torah. It was built and reworked by rabbis, mystics, and poets over a long period of time to express their ideas about the relationship between man and G-d.