Will Awakening let me tap all my lands and get them immediately untapped?
You could use Awakening to tap your lands twice in one turn, but not exactly in the way you expect.

Starting with your untap step, play proceeds as follows:


Your untap step starts, and all of your permanents untap.
Your upkeep step starts, and Awakening's triggered ability goes on the stack. You gain priority. This is your first opportunity to do anything.
You can now tap your permanents to add mana to your mana pool.
Awakening's ability resolves, and all of your lands and creatures untap.