Hey Adete,
There are few offbeat places in Himachal that you can travel to on a long weekend. Tirthan - Jibhi used to come first to my mind but then it is getting crowded and crowded every year. You can easily do a trip to Tirthan / Jibhi in about 4000 - 8000 depending on your stays.
Next on the list is Rohru - Kharapathar region, about 120 KMs from Shimla. Cam besides the river and you can save quite a few bucks. Can easily be done from Delhi within 3000-4000 with your own camps.
Another one I will suggest is Barot region. Very pocket friendly, offers options of camping and small homestays too. Again you can easily do it within Rs 3000-4000 over a long weekend. I have posted an itinerary as well here on Tripoto, can follow that :)
Then, when you travel in Uttarakhand, I can recommend Chaukori, Binsar, Kausani, Gwaldam as options along with Khirsu (https://devilonwheels.com/khirsu-detailed-travel-guide/).
All these places will provide you an experience amidst nature. You can check their guides on my blog if more information is needed.