1. I intend to purchase a new computer but am unsure as to what specifications (I do NOT want to break the bank!) would be good enough to run Microsoft Flight Simulator smoothly.
Two versions of MSFS are currently available: FS 2004 (also called FS9) and FSX. FSX is the most recent version, FS 2004 preceded it. FS 2004 will perform quite well on just about any recent PC. FSX requires a high-end PC to provide acceptable performance.
FS 2004 is slightly cheaper than FSX, and there are more third-party add-ons available for it (which is important if you want to get maximum realism out of the simulator). FSX has more eye candy built in (if you have the computer horsepower for it), and the internal structure is more amenable to the use of complex add-ons, although the add-on market still hasn't ramped up to the level of FS 2004 add-ons. Both use the same flight dynamics, which are largely table-based … meaning that realism is fairly good for real-world-based aircraft in normal flight, but simulation of unusual flight regimes is much more variable.
2. Recommended joystick/yolk/rudder pedals? Price? (Would like something relatively high standard because I will be training to fly to a Commercial standard later this year.
I like the Saitek X52 joystick and throttle quadrant. If you're willing to dedicate more money and effort to it, the Saitek and CH Products yokes are popular. Rudder pedals are a must, too (I use Saitek rudder pedals).
3. Which Flight Simulator? 'FX' perhaps?
I prefer FS 2004, but there are just as many people who prefer FSX. It depends to some extent on the add-ons you want (you'll want hyperrealistic add-ons for serious simulation) and the performance of your computer. Microsoft has stopped development on MSFS, so the FSX version will be the current version for the foreseeable future, and over time the add-on market for FSX will over take the older FS 2004 add-on market. Since they both use the same flight dynamics, the realism of flight in the sims is the same for both versions, although FSX has a few extra effects (most of them purely visual candy).
You'll definitely want to look into add-on aircraft, which are often so superior to the built-in aircraft in MSFS that it's like flying a whole new simulator. The big names are PMDG, Level-D, Eaglesoft, Dreamfleet, Carenado, Willco, etc. Some companies specialize in airliners (PMDG and Level-D are legendary for these), and others specialize in smaller aircraft (Dreamfleet has fabulous Beechcraft add-ons, and Carenado is known for its Cessna add-ons). Some are in between … Eaglesoft has a great Citation X, for example.
Other things you'll want to consider are membership in VATSIM or IVAO, so that you can use voice communications with ATC on these free virtual flying networks (some of the controllers on these networks are also controllers in real life, ditto for pilots). They work with MSFS thanks to a free plug-in. Add-on scenery is nice for major airports. Add-on weather packages such as Active Sky produce weather so realistic that it's hard to distinguish the simulator sky from a real sky. And many add-on bizjets and airliners include a FMS for which you may need regular database updates (about $20 a year). RealityXP produces gauges of such superlative realism that you can operate them using the real-world manuals (in the case of Garmin gear, the simulator code is actually written by Garmin).