Civilization – How is embarked movement increased

Seeing what units can still move

You can see whether a unit can still move from the unit list - you can open the unit list screen by clicking on the symbol and the top-left corner and selecting the 3rd option. Each unit has a colored circle next to it, as well as text explaining what it is currently doing (e.g. fortified). If there's no text, it means the unit is awaiting orders.

• If the colored circle is green, it means the unit has all its movement points remaining.
• If the colored circle is yellow, it means the unit has some of its movement points remaining.
• If the colored circle is empty, it means the unit has no more movement points remaining.

So in short, any unit you see with a green or yellow circle can still move this round, even if it's not awaiting orders - you can click it directly in that unit list screen to select it, or even double-click it to select it and center your view on it.

See all available tiles for movement

To see all the available tiles a unit can move to, either click the move-to button and then select a tile, or hold the right mouse button (don't just click it!). Then, when you release it, the unit will move to the designated spot. In both cases, you will have a blue outline showing the limit of where the unit can reach in the current turn.

If you use either of these methods and then designate a location beyond your movement range, you can even see the entire planned route (well, at least the first 8 turns of it or so), how many turns it will take to get there, and what position the unit will be in at the end of each turn.

Grey Emblem

I see you mentioned the unit emblem turning grey-ish. This only happens when a unit has no more movement points remaining in that turn. A unit may be busy doing something (e.g. fortified or constructing something) so it's not awaiting orders, but still with moves remaining, so it will still have a non-grey emblem.

I did a quick test using a couple different saves I had available to me, and found the following:

Route                   |  Cost (Movement points)
Road                    |   1/2
Road, with Machinery    |   1/3
Railroad for fast unit  |   1/3
Railroad for slow unit  |  3/10

More info

• Each unit lists how many movement points it has remaining in the bottom left when it is selected. The movement points are normally shown as rounded whole numbers only, but you can see the decimal number if you mouse over it.
• What terrain a road/railroad is on makes no difference. Roads on open terrain, hills, and forest all cost the same to move across.
• I tested both slow (2 movement point) and fast (4 movement point) units on each type f route. Railroad was the only one that showed a difference for different units.
• 3/10 (0.3) is an odd number, but its actually what I found.
• Going from road to rail or rail to road costs the slower 1/3.
• Remember roads do not bridge rivers until you get construction.

Edit: In my first test, I had an extra movement point on my units due to playing as Persia in a golden age. (I did not remember where it came from, thanks bwarner) I've tested again and got the same results.

Edit2: I incorrectly listed the tech that increase road movement speed as Construction. Construction is important here because it builds bridges, but the one that affects the speed of roads is Machinery.