Auli, Manali and Shimla is the best option. These are close to Delhi so, you can save a lot on transportation. You can also find snow there and some cheap stay options.
You can visit Manali or beyond it there are many beautiful offbeat destinations like Malana, Tosh, Grahan and even Kasol. Other than that Auli and Jhibi are also great destinations for a small budget.
The best place and a place nearest to Delhi would be Shimla, Kullu Manali. Other than that you can travel to Sikkim as well to witness the snow clad mountains. You can also check for Leh Ladakh.