I understand the yearly Torah reading, but when is the rest of the Tanach read?. is there a schedule for the rest of the Tanach?
As noted in the question, the torah is divided up into portions and read in its entirety over time (either one year or three years, depending on local practice). This is not true for the rest of the Tanakh; while parts are read on a regular basis, other parts are never read in synagogue (though they of course are studied in other contexts).
Each weekly torah portion is accompanied by a haftarah reading, excerpted from nevi'im (prophets). You can see a complete list of these readings at Judaism 101. Also noted on that page are special holiday readings.
The entire book of Esther is read on Purim, and the entire book of Lamentations (Eicha) is read on Tisha b'Av.
Ashkenazim (but generally not S'fardim or Chassidim) read the following books in their entirety on festivals:
- Pesach: Shir HaShirim (Song of Songs)
- Shavu'ot: Ruth
- Sukkot: Kohelet (Ecclesiastes)
Finally, some other Tanakh texts, most notably from Psalms, are used throughout the liturgy.
There actually is a schedule, though nowadays it is not very well known. A full listing of the where each section in Nakh (the Prophets and Writings) begins can be found here. These sections, known as סדרים sedarim, are marked in old manuscripts of Tanakh with the letter ס. There are 295 such sections in the link above, leading many to beleive that this division was part of an annual daily study cycle (leaving off the days of Shabbat and Yom Tov, one is left with approximately 295 days in a regular Jewish year). I will note that there are some variant traditions about exactly how to break up the sections in certain places. The popular Koren Tanakh includes these breaks (and numbers them) for use in study and reference.
As for ritual public readings, Monica's answer gives a nice summary of the main ones (Torah reading on Shabbat morning, Haftara reading on Shabbat morning, and Esther on Purim), but I'll point out two others here:
Many Sefardim and Yemenite Jews have the custom to read aloud the book of Job (Iyov) on the Ninth of Av.
The Talmud (Shabbat 24a) records a practice to have an additional Haftarah reading on Shabbat afternoon. It is disputed (see Rashi and Tosfot ad loc.) if this was a reading from the Prophets or the Writings. It is no longer practiced today.