Wasn't really sure whether or not to pose this as a question because it's a little vague because I have so little idea what to do.

After consulting a different GOG topic to even get the game to run, now I can't figure out how to do anything and the opening cutscene told me approximately nothing.

I began the game and had a rather polite conversation with a guard (seriously, I can't recall the last time I saw common courtesy of this level in a game) and then wandered around really slowly and found myself in a shop where I can't afford anything and a rat-infested storage building (which can only be seen from a rather strange viewpoint, you can't walk directly towards it to see it... case of Resident Evil-style fixed camera angles) where I found myself trying to open a chest with no instruction whatsoever. I made the mistake of trying to 'probe' the chest (which seemed like the obvious thing to do, like scouting before rushing into a fight) and then holes in the chest slowly filled up with things and then I got hit...

Long story short: I have no idea what I'm doing, I don't know where to go, I walk rather slowly and the camera angles aren't helping.

Admittedly about 100% of the dozen or so games I've played in the last two weeks haven't been helping my patience, but this is being even less straightforward than usual.

Can someone direct me to a place that will explain how to get things moving in early Return to Krondor or simply tell me where to go when I first start up?

Also, I seem to be unable to save my game, which isn't a very immediate problem (since I can't figure out how to do anything) but I imagine will be slightly more important once I do get going...

AFAIK, the manual should explain the lockpicking minigame. How you do it will depend a bit on whether you chose reflex-based or skill-based lockpicking in the "new game" screen.

Camera angles are generally fixed, but in some cases the ] and [ key will allow you to rotate through several. There is a little icon on the bottom bar that appears when this is possible. (Most generic building interiors don't have this, but specific event/combat ones may.)

As for the first thing to do: you need to go to the city map screen and visit the gates to meet up with the new court mage. This map can (IIRC) be accessed from the game menu - although maybe not in the intro - but you can also leave the current area by walking past the zone marker - an upright stone which, in the case of the palace district, is next to the shop.