Where is the earliest place to sleep?

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.

when you first enter the keep at helgen there is a bed, if you side with imperial the bed(s) are in front of you, if you side with stormcloaks, then after you kill the imperials you can go through the gate to the same room with the beds.

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