I hiked to Lago di Sorapis last month. There’s tons of ‘Rifugio’ (Hostel like accommodations) all around. I stayed at Rifugio Vandelli and hiked train #215 to Lago do Sorapis :-)
Like Winsie suggested, check the different Rifugio (mountain huts with basic facilities) I personally suggest Rifugio Auronzo near Tre Cime and Hotel Montana in Cortina. Cortina could be a good base to go around.