It would be more economical and convenient to reach Manali from Delhi and continue on to Leh from there.
You need to be purchased 1-2 days before your journey and you need to board the bus from the main Manali Bus Station. These go via the Rohtang Pass, Kunzum Pass, and then to Kaza. You can easily reach Spiti Valley by taking a bus from either Shimla or Manali. There are around three buses that leave for Reckong Peo in the evening and reach by morning. The same bus takes a halt and picks up more passengers and leaves to reach Kaza by the evening of the same day.