Where is the earliest place to sleep?

Solution 1:

There are 2 places you can sleep before reaching Riverwood.

• A bed inside the Embershard mine (as stated by Rodrigo Guedes)
• A hunters camp at the first 3 ritual stones/standing stones.

Both will give you the "Rested" state.
Note that the beds inside the mine will require you to kill 3 bandits before being able to reach it.

Screenshots for locations:

The First Standing Stones

The Hunters Camp From The Stones

The Bed Providing The Rested State

The Camp Site Inside The Embershard Mine

Solution 2:

If you want the "well rested" status, the earliest place is the inn in Riverwood. If you don't, there is a place to sleep in the mine before you reach Riverwood.