Hey,
From Madrid to Segovia, the actual pilgrim services are few, but there are a couple of places to sleep in each town. After Segovia: one day - one town - one Albergue. All rustic but good.
TRES CANTOS: Hotel VP Jardin, COLMENAR VIEJO: Gran Hostal El Chiscon, MANZARNERES EL REAL: Hostal La Pedriza, CERCEDILLA: Hostal Aribel Longinos. It was a few years ago, but none of them were over 40 euros.
The walk to Segovia is long and hard (one steep elevation up the roman road), might want to think about the detour to La Graja de San Ildefonso (a few nice, reasonably priced hostals and such, and there's a Parador too, of course not reasonably priced, but cheap for a Parador). The walk from La Graja is 12k on a side, nice walk. I also recommend Hostal Forno in Segovia, family run, clean, nice little room right by Plaza Mayor. Have Fun, Bob