Don't put too much emphasis on the bass note - it's a helpful tool, but not an infallible guide.
Many, many chords are played with a note other than their root note played as the bass.
The Em vs C is something different - it's only a semitone, but it's an important semitone. You should really work on getting a feel for a minor chord vs. a major chord. The easiest way is to play something like E, then Em, then E, then Em - and then A, then Am, etc... until you can "feel" the difference.
Once you have that, you should usually be able to tell at least whether a chord is major or minor - even if you don't know anything else about it (there are special cases that will still trip you up from time to time).
If you can determine what key the song is in, that will help you a lot. I'll give an example from a key you're not familiar with (so you can work out the ones you are for yourself

). If I know a song is in Eb, then I'm going to expect Gm, Cm, Ab, and Bb to be played (quite possibly in that order - look at the scale degrees).
It really is a lot like solving a puzzle - the key of the piece, the melody note, the bass notes, and the chords surrounding it are all clues to what the chord might be, but you may still have a bit of work to do to puzzle out exactly what chord it is.
An extreme example would be "A Horse With No Name" by America. It's a simple two chord song (probably the easiest 'real' song to play there is). One of the chords is Em. I'll wager there are few people here who could puzzle out what that second chord is though - at least not without a lot of work (maybe even using an equalizer to isolate notes).