Background to this source sheet:
Trumah-is given to the Kohein.
Ma'aser Rishon-given to the Levi during the first six years of the Shemittah cycle.
Ma'aser Sheini involves bringing the Ma'aser food to Jerusalem and eating it there. It is offered ON TOP of Trumah and Ma'aser Rishon in years 1, 2, 4, and 5 of the Shemittah cycle.
Ma'aser Ani is given to the poor. It is offered ON TOP of Terumah and Ma'aser Rishon during years 3 and 6 of the Shemittah cycle.
Orlah-the fruits of a tree are prohibited for the first three years of the tree.
חנטה refers to blossoming of fruit trees or when the fruits form a distinct shape. You will see various forms of this word throughout this source sheet.
The following source is taken from Rabbi Ovadia Bartenura's commentary to the Mishnah:
The following source offers the Gemara's reasoning for the date of the first of Shevat, though it applies to the 15th of Shevat as well. Studying Rashi in source 4 will greatly help you in figuring out the passage. If you still cannot figure it out, you are welcome to look up its translation on Sefaria, which now contains the Steinsaltz English translation.
The following passage of the Jerusalem Talmud also discusses the reasoning for the 1st or the 15th of Shevat as the cutoff date. Although one explanation is the same as that of the Bavli, it presents another opinion as well. Since the language is opaque, don't worry about skipping this source if you can't figure it out.
In the following source, Maimonides gives a definition of when fruit is ready for Ma'aser.
In the following source, Maimonides explains how Tu BeShvat is relevant towards Ma'aser. His teaching is somewhat similar to the commentary of Rabbi Ovadia Bartenura.
The following source does not relate to Halakhah, but it beautifully illustrates a connection between fruit trees and the need to care for the generations that come after us. You might have heard about Choni Hamagel, Choni the circle drawer, from another legend about him. He famously draws a circle around himself and refuses to leave it until God brings rain. The story contains a lot of Aramaic, which is why I included the English translation in this source.