Drag and drop
The easiest way is to drag and drop the troops on the map. However, it will only work if all units are of the same type (land/naval/air). You can also drag separate units or unit groups from the sliding out info menu (hover on the troop icon).
Move troops menu
To select which units you want to move, click 'move units' button in the city or click the troops on the map. In the appeared menu you'll see 3 tabs for different unit type (each type has to be moved separately!) and sliders for each individual unit. Click on the numbers to the left and to the right of the slider to select all/none or move the slider itself. There's also an option to select all/none units of in the tab, below the sliders. After selecting the units, click the green 'move' button in the lower right.
Unit image in the City menu
Clicking the unit image in the city menu will move all units of the kind (all Tanks, all Infantry etc.)
There are two transports in the game: Air Transport and Sea Transport (there are also Submarines, which only carry Marines). To load land units into them, go to the Move troops menu, select units from the 'land' tab - now, if you have any transports in the units stack, you will see additional buttons below - 'load to Air Transport' and 'load to Transport'. Note that loaded units cannot participate in battles when attacked, and will perish if the transport is destroyed.
To attack, simply move your troops onto your enemy's. You cannot attack your allies - first you need to break the alliance (in the Players menu) - this takes one turn.
You can undo the last movement by hovering the mouse on the movement path and clicking the appeared 'x' button. If the button doesn't appear, then the undo is already impossible.
All units have certain movement range. You can keep moving your units until it expires, so your movement doesn't have to be straight and can consist of several paths. If units in the group have different ranges, the group's range will be the average of them. You can turn on/off the movement range for all units under the minimap - this can be handy to show you the units that haven't moved yet.
Likewise, all units have visibility range (it can also be turned on/off under the minimap). If enemy troops are beyond your visibility range, you won't be able to see their numbers and unit details.