I've been using an XE1 for two years now. Bought it with the Fuji 14mm, and 1.5 years ago added the Fuji 23mm.
Unlike Anthony, I strongly prefer the left side finder of the rangefinder style. That's why, instead of getting the XT1, I'm waiting to see if Fuji will continue evolving/improving the rangefinder style body.
The picture/file quality is good with the XE1, no problems there. Photo Ninja does a better job than Adobe, although for small prints/displays and certain subjects with larger displays/prints, the Adobe raw conversion can be fine.
The XE1 is slower, in terms of being able to quickly, repeatedly press the shutter and get it to fire, than the newer XT1. This would be my main gripe with the XE1. Also note that due to focus-by-wire with the Fuji lenses, the lens focus is lost whenever the camera sleeps, or is cycled off/on.
Just recently, my XE1 has acquired two dust particles beneath the sensor cover glass, one of which shows in the sky area in a conventional horizontal landscape composition at f/8 or smaller.
I recommend the XE1 as the lowest cost way to get the Fujifilm X-Trans 16 megapixel sensor and it's charming image quality.