I know how to select faces, vetices and edges - what I don’t know how to do is get back into Object selection mode.
Right now I’m in face selection mode - but I want to grab and move the whole object - without having to shift select all of the faces.
How do I get into Object selection mode?

but I want to grab and move the whole object - without having to shift select all of the faces.
You could press A to toggle between select all and deselect all selected or box select all objects (B)

How do I get into Object selection mode?
Select Object mode from the 3d view header or press Tab key to toggle between object and edit mode.

Selection in the manual

It is important to remember the difference between moving something in edit mode and in object mode though.

To summarize the difference:

Moving stuff in object mode will move the origin where it is in relation to the object, but will alter the objects data in the properties panel (press ‘N’ to see that panel). If you scale for example, it will alter the scale of the object away from the 1-1-1 ratio it starts with This is important, because that directly effects modifiers and how they play with your model, the size of items in edit mode, and other important things. As a general rule, you want to try to keep the scale, rotation, and location all it’s default values by applying them once changes are made. To apply them, press Ctrl + A in object mode.

If you press A to select all as was suggested in the above post and move all of the verts in edit mode (or object mode, A to select/deselect all works in both modes), the scale, rotation, and location will remain whatever value they were in object mode, but your origin will now not be the same. It will remain where it is, regardless of where you move your verts. This can be undesirable, as many things can revolve around the location of the origin with respect o the object, such as modifiers and what axis your object rotates on. There of course, are times when you want the origin to move. Two easy ways to reset the origin of your item, or place it at like the 3D cursor for example is to press: Ctrl + Shift + Alt + C, or in object mode click the origin button under the transforms panel.

There are of course other ways of selection besides just A to select all, lasso tool, box select, linked select, edge loop select, ect. There’s lots of other good moths for selecting you can look at, but I suggest the five I listed as the most used and useful methods to select.