When do units move at the beginning? 

It's the same system used in the Civ and GalCiv games.

I think I just was confused and got it wrong how the units are moving. Either that or it changed in 1.07.

Units do move at the end of the turn.

Not sure why OP thinks they don't.

I agree with you, but also with the OP's observation that the current system is a little off-putting.  The whole thing as is just looks a bit awkward, since I don't see enough visual cues that my unit is really going to move.  Once I've finished with my active moves, I'm ready for the end turn button, but since my pre-programmed moves haven't been executed yet, I keep wondering if the remaining units really have orders or not, and often end up clicking on them to double-check or even redo their move order, which defeats a lot of the reason for setting a long move to begin with.  I realize some of this is just me learning to adapt to the engine, and unlearning habits from other games (at least Civ has a "Contiune move" button), but if other people are getting bothered by this too, maybe the interface needs a change to something a bit more intuitive.

Bingo.  Civ 4, you get a nice line, showing where every move will stop, and how many turns in all.  This is enough to reassure me that the game has understood my instructions and is showing me the path it intends to take, further reassuring me that the game will indeed accomplish the task I set it.  If I should worry about it, another click on the unit shows the path again.  Here, there is no path shown, just the destination square.  If I click again on that unit, and the destination square is not on the screen, there is nothing telling me the unit is on a long move.

Just showing the intended path/number of turns would probably go a long way to clearing the move thing up.  Still, this is a basic usability issue that should have been addressed a long time ago.

After further review, the issue I was thinking of in my original post is definitely still happening in tactical battles.

Or they can just add a Brief text that says Turn ## when the turn official starts. So you hit end turn, everything moves, Turn ## pops up on screen and you go about your turn. Minimum coding and implementation needed, so fast, easy, and cheap. Which allows more important things to be addressed.

I'd like to see something on the main map like in tactical battles where it shows all the squares where a selected unit could move for the remainder of the turn--or at least show the number of moves left in the turn over the head of the unit, or something.

Also, if a unit has a destination tile set, I'd like to see the target desination more clearly, like make there square flash or have a big flag on it, and an arrows leading from the current square occupied to the target square.

Finally, say we want a newly produced unit to join our sovereign's army.  It would be great if we could select a unit, rather than a square, as the destination, by control-clicking or something like that.  This could be enemy units too.