I'd like to view the progress of several badges, specifically the 'Sportsmanship' badge. Is there a way I can see how far I've gotten, and/or how much I have left to obtain a specific badge?
For most badges, no, there's not a built-in way to view progress. There are a few exceptions to this, like the review badges, which you can see if you hover over progress bar at the top of a specific queue, but usually badges don't have anything.
People have made some queries over on http://data.stackexchange.com which give you a rough idea though. Here's a query for Sportsmanship, but it's not totally accurate. For one, your votes aren't public, so the best people can do is show you the total number of possible votes, which would usually be quite a bit larger than your actual votes. Another thing to consider is that data dumps only happen about once a month, so it will usually be kind of outdated. But getting something like "74 rows returned" will at least tell you that you could have 74 out of 100 votes within the last month, which is better than nothing.