How do you find the root of a word?

There is no hard and fast rule for this -- it takes practice and familiarity with prefixes and suffixes. That comes over time -- lots of reading, vocabulary tests, and looking words up in the dictionary or on online sites, if you prefer those. After a while, you can analyze a word, separating out the prefixes and suffixes and zeroing in on the root.

For example, if you see the word "ursine," you should know that "ine" is a common suffix and that the root is "urs." It comes from Latin "ursus" for "bear," so the word means "of or having to do with bears," or "bear-like."

I repeat, practice is all.

You can also cheat until you get better at it :D
There is an online etymology dictionary at:

http://www.etymonline.com/index.php

Example from the site:
etymology
1398, from Gk. etymologia, from etymon "true sense" (neut. of etymos "true," related to eteos "true") + logos "word." In classical times, of meanings; later, of histories. Latinized by Cicero as veriloquium.

I'm not sure if you mean etymological roots here or not, but they are found in the same way you find English roots.

Example: the English root word of "unfailingly" is "fail."

It's pretty hard to FIND an etymological root if you don't KNOW basic Latin and Greek roots, as well as French, Anglo-Saxon, etc. So you would want to familiarize yourself with those roots. Any good bookstore will have resources for you.

Surely the first place to look (if you can't simply work it out) is an etymological dictionary, whether on line or an actual book.
For instance I have on my shelf behind me "The Concise Dictionary of English Etymology" by Walter W Skeat. Many dictionaries like the Oxford Dictionary also give the derivation as part of the definition of a word.