Don't feel bad tom. Autos can be complex pieces of machinery and without pulling one down can be difficult to understand. As with all machines breaking them down into subsystems helps to conceptually grasp how they operate. They're not all that hard though. Lots of places on the Net to help you understand.
Your symptoms could be caused by many things but a glance at my A340E book diagnostic chart shows they could be caused by incorrect line pressure or an incorrectly adjusted throttle cable. Shift shock is usually associated with line pressure problems so have it checked and be sure the throttle cable is adjusted correctly
Nick is right: Before you do anything start with the fluid. The fluid not only "drives" the transmission but also cools and lubricates it. An automatic transmission is above all a hydraulic system. The secret to increasing the longevity of lubricating fluid and the system it's used in is to keep it clean and cool. This is especially true in hydraulic systems. Contaminated fluid and heat are the single greatest killers of them. The situation is made worse in cars because as the clutches and brakes operate they shed friction material into the fluid the same way a manual clutch sheds it's friction material over time. The heat problem stems from an auto tranny being 10 pounds of stuff in a 5 pound bag.
Many times I've seen great improvement after flushing and fluid replacement. Course, it won't fix worn out parts. It may be too late but try it. You can't simply drain the pan because you'll only get 30% or so of the fluid out. If you do decide to go that way you should let the transmission drain overnight and do it 3 or 4 times over the course of a few days. Be sure to move the shift lever through every position to flush the entire valve body after each fill. You should also drop the pan and replace the filter if there is one. Not sure about the 340 but since it's an older model I'm guessing it has a screen in the pan. Also, if there's a magnet in there be sure to clean it.
The best way to replace the fluid is to take it to a shop that has a tranny flushing machine like a T-Tech. Or you can do it yourself. Disconnect one of the cooler lines and put a hose from it into a bucket and start the engine. Shut it off when the hose begins to spit air and refill with the same amount of new fluid. You'll have to do that several times until the fluid color changes or you've replaced the specified fluid capacity of the transmission. For a good flush run 1.5 times the specified capacity through it.
This may solve your problem but it's a crap shoot. That's why it'd be risky to use expensive synthetic fluid. You could try using non-synthetic and if the problem clears up do the process again with synthetic and call it a day. If a fluid change doesn't help you'll need to get the tranny looked at by a professional.
Lastly, think about installing a better cooler and for sure install a better filter....especially if you have the tranny rebuilt. The stock screens used in most trannys are jokes. There are inline filters available but you can always buy a filter mount, plumb it into one of the color lines, and use a good spin on oil filter. You can also use a real hydraulic filter. Imho the best setup of all and the one I use is on all my autos is to install a depth (bypass) filter. That said, a spin on oil filter will work great and beats the factory gravel strainer by light years. Change the filter every 40K miles.
If you're smart you'll also put a filter on your PS system. A decent one is the Magnefine. It's normally sold for trannys but I feel it's somewhat small for that purpose, at least for an initial clean up. Great for PS though: http://tinyurl.com/yed4wx
Course, you could drive the crap out of the car without doing squat to make it last, all while investing in oil of snake and other worthless products rather than proven maintenance principles. While the Supra community seems to be full of such people me thinks you can do better.
Anyway, good luck with this and let us know what you find.