I deal with classical music in various ways on a regular basis, so I thought I'd chime in. Assuming we talk about stuff written before, say, 1900, the music itself is public domain. But any performance, unless specifically stated otherwise by the performer(s), is property of the performer(s) and you need permission to use it.
Even the files on Wikipedia Commons, linked above, are provided with a license. So when you use them, take care to read the license to figure out what you may and may not do with the recording. One thing you're almost certainly required to do is to credit the performers.