Hi Vidhi, You both can easily spend 3 days under Rs. 5000. You can go for boating in Nainital Lake, shop at the Mall Road, and enjoy views at the Snow view point. You can find recommended place to stay and a guide to Nainital attached below. Keep exploring!
Most people have already mentioned the main places. There is the Bhutia/Tibetan market too which is much cheaper than Mall Road. You will find all kinds of shops and things there. There is a restaurant there called China Valley where you get amazing Chinese Food, especially the momos, and at a cheap price too! At the Mall Road, Machan, Zooby's Kitchen and Pots and Stones Cafe are very good food joints but they're a bit pricey. When we go to Nainital we usual stay at the Kumaon Mandal Guest House. You get free breakfast too. However, we usually go there during the off season so we get the hotels pretty cheap. I don't know what the price will be during season time.