Often I find myself in the middle of some part of davening when the chazzan starts saying kaddish. Should I interrupt to answer or does it depend on which part of davening I am holding?
There are different rules depending on which part of Davening you are up to, and the rules also depend on which part of Kaddish is being said.
During preliminary parts of Davening (Berachoth, Korbanoth, Pesukei DeZimra, etc.):
During Keriath Shema' UVirkotheha:
Interrupt between paragraphs (this is a technical term, not determined by the page layout of your Siddur) for:
Interrupt between verses (still technical, but a lot more intuitive than "between paragraphs") for:
During Shemoneh 'Esreh:
During the actual 'Amidah:
During Elokai Netzor:
After Yihyu LeRatzon (at the end of Elokai Netzor):
SOURCES: Memory of the rules printed in Artscroll and Siddur Tefillath Yosef, as well as Jewish day school instruction from years ago. I believe their source is primarily Mishnah Berurah, but see the M"B itself for more details and slight differences of opinion brought by the Magen Avraham and Peri Megadim. Also see the Shu"'A and RaM"A for slight differences. First see O"C 66:3 and M"B there for the rules of interrupting K"S, then see O"C 122:1 and M"B 4 who equates Elokai Netzor to K"S. M"B on 66:3 also give a few Kol SheKeins about "Tahanunim" and Pesukei DeZimra.
Both Sidur Koren and Sidur Tefilas Kol Pe have a table that shows exactly where one should or should not interrupt for Kaddish. I will attempt to summarize later or see if I can find the table online.
Rabbi Mordechai Eliyahu, in the appendix to his edition of Kitzur Shulchan Aruch, prints a table for when this is permitted (Ashkenazi one from A. Bloom publishers, Sefardi one by himself). Remarkably, it is available online (p. 685 of this pdf).