Open a browser window and head to the etymology reference section on Dictionary.com (dictionary.reference.com/etymology). The etymology section works differently than the regular, search-friendly dictionary. It's a bit more like a hard-copy version: You need to flip through pages and browse for your word. The list on Dictionary.com only includes words for which a historical root has been identified by a scholar.
Click the first letter of the word for which you are looking for the meaning and history. Say you've just encountered the word "tawny," and that you have an understanding of its meaning but not a full idea. The Etymology Dictionary is sorted into pages like a proper, paper dictionary, with a group of words on one page. The range is defined. Click "T," then look for the group of words that contains "taw." Click "taskless... tay-sachs," since "tawny" will be between these entries.
Look through the list of words and click "tawny" to find its etymology. You can see the definition (tan-colored) but also more information than you might in a regular dictionary. On this page, you learn that the word dates from the 14th century Anglo-French "tauné," meaning related to the brownish-yellow of tanned leather. If you can't find the word you are looking for in Dictionary.com's etymology section, try looking through Etymologically Speaking (westegg.com/etymology) or the Online Etymology Dictionary (etymonline.com).
Read up. Beyond looking up each word on an individual basis, you can build a bit of an etymological dictionary in your head. Argot Language Center (r-go.ca) has an Etymology resource for ESL-learners that covers the basics of prefixes, but explains the origins and meanings of 40 or so common parts of English words.
Stay relevant. Language is always evolving, both forward and backward. New technologies require new words to describe them, and words enter vernacular as they become used more regularly. Historians are also always discovering texts and documents that shed new light on the history of language. The best way to keep tabs on the evolution is to read a blog like WordOrigins.org, which updates semi-regularly and features news summaries and academic writing about the history of words as well as a huge list of words.
Invest in your learning. The tome of etymology is Oxford Dictionary of English Etymology, a print-only resource that weights almost four pounds. It's more than a thousand pages of detailed, scientifically presented word meanings and histories. Beyond the Oxford etymological dictionary, look for colleges and universities in your area offering an etymology class. These classes fall under the heading of language and linguistics, so check with these schools for more information. A one-semester study of etymology will place you among the experts, at least within your circle of friends.