Hi Katie. I've got an X-E1 but from everything I've read, I understand the RAW files are the same as from an X-Pro1. There is a lot of discussion in forums and blogs, along with examples. I recommend you search for "X-pro1 raw" or, to catch more recent postings, search for "X-E1 raw".
The software that comes with the cameras does a good job of processing RAW, but it's widely regarded as having a bad interface. The retail version of the software (Silkypix) is expensive, with no upgrade pricing. It's not a DAM solution like LR.
Phase One has a beta of their Capture One Pro 7.0.2 that handles X-Pro1 RAW. It's been well received (almost raved about). You have to ask for the beta. It expires 31st Jan, so hopefully it will be released before then. Capture One Pro is also expensive, but from version 7 does the full workflow.
In my book, LR 4.3 doesn't do too bad a job of handling RAW. It's certainly usable. Everyone expects that Adobe is working to improve RAW processing, and the Fuji Guys have tweeted that Adobe has the necessary documentation to do a better job.
Until LR improves you can either accept the results, or process the files in the free Silkypix converter, and import the TIFFs into LR. I'm mostly taking RAW+Fine jpg so am covering my bets. When shooting RAW you will want a fast card. I use SanDisk Extreme Pro, and it does make a big difference. Sometimes I take sets of 3 jpegs using the "Film simulation" bracketing. The jpegs do look nice, but that form of bracketing doesn't allow you to include a RAW.
Overall, I'd recommend you take RAW+Fine and continue using LR 4.3. If you find LR's RAW conversion doesn't meet your needs, export from the free converter. You may even find you're happy with using the jpegs.