I get the cover off the shift linkage at the tranny and am greeted with a bath of fluid that I assume is trasmission fluid...Is this normal, or a sign of something wrong???

It's a sign that the shift console seal has failed and is allowing fluid to leak past. It's pretty easy to replace especially if you have Dr. Evil's video. Better check your fluid level ASAP.

You got a leak, check the level asap via the fill plug on the side of the tranny case.

if low, fill it or best yet warm it up by driving on the hiway for a good deal (just like you warm up engine oil) then drain it right after stopping, so as the most bad stuff gets drained out.

make sure you aint running low. once drained is a nice time to replace the seal, or if you cant wait, just top it off for now and monitor leakage, could be a very slow leak. just dont run low dont run low, dont run low!!!

New to this whole thing...Is that something that can be fixed while in the car, or will I have to drop the tranny?

It can be fixed in the car, but the transmission oil must be drained. Once the trans is empty, remove the shift linkage, shift in to neutral, then undo the two nuts. Pull the whole shift console down and then you can work on it on the bench. Not a hard project.