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.
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.