I'm coming across a number of scenarios where I'm being targeted by enemy shuttles and transports... and the computer won't target the threats! No matter how many times I press 'R' my screen is left blank while the enemies just shoot the hell out of my ship until I'm destroyed. How do you target enemy shuttles and transports?!?!

"R" is usually just for fighters, most shuttles and transports may be excluded depending on the game (which one are you playing and which version?).
You will have to target them manually using T and Y to scroll the targets or use button 2 for selecting the target in your sights.

I believe the E key may target the nearest enemy attacking you, while the R key just targets the nearest enemy.

R targets nearest enemy fighter.
E targets nearest enemy anything attacking you.

Okay I pressed 'E' many times and ended up ramming into an enemy transport without ever targeting the unit.

Seriously how do you actually target enemy units? I couldn't give a damn how NOT to target them.

"E" works only for fighters too.

Any ship that isn't a fighter or saved in the presets (shift + relative F-keys to assign, they vary between games, IIRC TF uses F5, F6 and F7) has to be targeted manually either by scrolling targets with the Y and T keys, if a main mission objective there is the O key in some games, or by putting it in your sights and tap joystick button 2 or the right mouse button.