For a small camera to take everywhere by a newbie I actually would recomment the Sony RX100 Mark 2 as the best available option.
I dont know if its inside in your budget, though.
Anyway its IMHO the best general camera out there, if you are a beginner and want stuff commonly popular among beginners, like a good focal length range. Theres a successor, but it actually had not too many improvements and the lens is brighter, but also offers less focal length range. The Mark III has an quite awful builtin EVF, but you can get a clickon EVF for the Mark 2 and its better in quality.
About the X20, well I dont own this camera so I can only repeat what others told me about it. It has a half as large sensor and the sensor tech is not as good as the Sonys, but it has phase AF pixels on the sensor (*), and personally I like the manual way you make the lens expand on the X10/X20/X30 cameras.
What I do NOT like about the X10/X20/X30 is that they are X-Trans, i.e. they produce a RAW format thats hard to properly process for general image manipulation programs like Lightroom. Theres no problem with the JPEGs of these cameras though, Fuji themselves do a really good job in processing their own exotic sensor output, they just dont tell others how they're doing it.
(*): That means you can have some sort of better performance with moving subjects in good enough light, because the camera knows if the focus is too far and has to move closer, or the other way around, while a standard contrast autofocus only tells the camera "well ... its not in focus just yet".