I know I can click on the "move" button
and use my mouse to drag the object around. How can I do the same using my keyboard?
You can use Q/W/E/R/T to trigger those buttons, W being "move", E "rotate" and R "scale". This is faster than clicking the buttons in my opinion.
However, there is unfortunately no key for "grabbing and moving" an object, like for example blender has.
There is no keyboard shortcuts for moving a selected object around (afaik). What I like to do is change the location on each axis by selecting the gameobject in question and moving it using the inspector; If you mouse over the axis you are working with (for instance: X), you can 'drag' it right or left to increase and decrease that value. This might be a good alternative.
I just made a Unity editor script to accomplish exactly what you are looking for. Check it out at this GitHub Gist
I don't have my copy of unity to hand right now to try it out, but I couldn't find any way to move gameobjects using the keyboard. However, I did find this page full of keyboard shortcuts. Also, if the issue was not having enough precision with the mouse, you can always use snapping to make sure that the vertices and objects align properly.