1. You could check your server logs or stats to see how often the feed files are accessed, but this is technically inaccurate since it displays how often the feed files are accessed, not how many unique visitors access the feeds, and it won’t filter out bots either.
2. If you politely asked all of your users to subscribe to your feed through Bloglines, you could keep track of how many Bloglines subscribers that your feed has, but this will still ignore non-Bloglines subscribers.
3. If you ran your feed through Feedburner, you would be able to see detailed statistics of your feeds. These statistics would count unique readers, display which aggregators they use, and filter out bots. This probably is the best solution, but it does require the most work. See this for more details: http://codex.wordpress.org/Using_FeedBurner
What “other readers” are you referring to? I have several feedburner feeds in my daily feed-reading list and they work perfectly in RSSOwl (desktop reader), PulpFiction (desktop reader), Bloglines (online reader), Newsgator (online reader), Google Reader (online reader), and Rojo (online reader).
FWIW, I’ve had no problems with Feedburner. As with statistics gathering with regular webpages, the tracking of unique subscribers will not be 100% accurate (consider how you track someone using bloglines or another webbased reader) but Feedburner appears pretty accurate as well as being easy to use.