The distance from Delhi to Spiti Valley is around 750 km. The travel by car might take 12-14 hours, depending on the weather and road conditions.
From Delhi, you may take a bus or a cab to Manali, which is around 550 km distant. From Manali, take a shared or private cab to Kaza, the major town in Spiti Valley.
You may take a bus or rail from Delhi to Shimla, then rent a cab or take the bus to Reckong Peo, from whence you can take a shared or private taxi to Kaza.
Another alternative is to travel from Delhi’s Indira Gandhi International Airport, which offers greater flight connections, and then take a cab or bus to Spiti Valley.
You can also take a flight to Bhuntar and then a cab to Spiti, or drive/take a bus from Delhi via Shimla and then to Spiti Valley, with an overnight stop at Reckong Peo for acclimatisation.