You van do hamta pass. its breathtakingly beautiful, moderate difficulty and well also economically reasonable to do. also you can stay in a hostel in manali and then continue to spiti after the trek. as spiti is quite economical as well. will suit your budget the best
Trekking is the best way to explore a place. 2 treks which I can suggest to do in India. 1) Sar pass it starts from kasol 13800 ft trek 8 days trek. including food approx cost 15000 max for whole trek fees excluding travel fares to kasol.
2) Chadar Trek starts from leLehit's a frozen river and frozen waterfall trek. a week trek including food 25000 max cost excluding travel fares to Leh. other treks also
chandratal trek hampta trek I guess
Auli, Gangtok, Tawang, Spiti Valley, Sangala, Gulmarg, Rishikesh
you can visit distant villages in Himalaya, trek in valleys, and will get to live in most amazing hostels or local stay. I would say don't go to commercialized places like Manali. Those are really crowded.
With that budget, you can go around treks near Pin Parvati, there are many. Kheerganga, Malana, Rasol, Pin Parvati, Pulga. They are on the same belt. Staying involves not much of a cost. If you have camping tools or can rent a camp, it would save you money. And Pin Parvati Valley gives you an amazing experience of mountains! All the best!