I planned a research paper discussing Cyrus the Great and his widely-renowned tolerance of other cultures and beliefs, but I'm having trouble finding sources that actually expound on this. I have a few secondary sources that are pretty good, Pierre Bryant's From Cyrus to Alexander and some others, but what primary sources back this up?

Check what primary sources your secondary sources are citing. Encyclopedias can also help if you have no idea where to start (in German we have Paulys Realencyclopädie and Der Neue Pauly, I don't know what their English equivalents would be).

Herodotus aside Xenophon would obviously be an extremely important source on Cyrus.

I've seen a few passages about Cyrus from Herodotus, but I don't actually have Histories in book form. Is there a searchable online version of Herodotus?

Perseus has searchable online versions in both English and Greek but it shouldn't be too hard to find a more recent edition in your university or department library.

Also... how the hell do I do an APA citation on the Cyrus Cylinder? I've just been able to find the text of it on a British Museum webpage, and I'm sure I'm not allowed in this 100-level history class to use online sources.

Wikipedia mentions some editions and translations. Maybe there also is a comprehensive collection of Akkadian inscriptions? I study Latin and when we have to cite inscriptions we do so from the Corpus Inscriptionum Latinarum which collects basically all ancient Latin descriptions ever found anywhere.