1. FD needs to be compatible with both your shifter and your chainrings. Shimano's FD-R440 is specifically made for combining flat bar shifters and 9sp chains with road double cranksets. It's equivalent to Tiagra in quality.
2. For shifters, you can get SL-R440 shifters which are also in the same "non-series" Shimano flat bar group. Or for maybe a few dollars less, you could get Deore (SL-M590). Both are good quality. If you want to save some money, Alivio (SL-M430) is a budget 9sp shifter. Alivio probably lacks the same trim features of ththe more costly shifters. Consult a shop or Shimano tech docs.
3. Microshift also offers a full selection of shifters and derailleurs for all uses. Although I haven't heard anything directly about their flat bar goods, what has been said of their road products has been generally quite positive. For your needs, Microshift offers TS83-9 shifters and a compatible FD-R42 front dérailleur. Their products are designed for Shimano compatibility, so your Sora RD will work fine with it.
All of the shifters mentioned are 2/3 x 9 compatible. That means they will operate either double or triple cranksets; it's up to the installer to set them up whichever way. I think all 9sp Shimano flat bar shifters you'll come across will be double/triple. It's not an issue, though. Just set them up so the two highest shifts will be in use, leaving the bottom shift as a dead shift.
4. Get a compatible bottom bracket.