I studied in Delhi, more than 15 years ago.
Delhi is a state not a city. Its massive. Distance from North to South is around 50km. New Delhi is the city.
Most embassies are in or near Chanakya Puri. Its also a safe place. Maybe hotels are a bit expensive but I suggest finding a place near Chanakya Puri.
Most students live in an area called Kamala Nagar/GTB Nagar/Model Town. These areas are also safe because foreign students live there. Kamala Nagar may be a bit cheaper than Chanakya Puri.
Shopping:
Unless you can speak Hindi like a local, they WILL rip you off. Usual shopping places are Sarojini Nagar, Connaught Place (Palika Bazaar), Pahar Ganj, Janpath, Karol Bagh, Chandni Chowk. Shops are everywhere in Delhi.
If you don't want to get ripped off, go to supermarkets. Unlike Sri Lankan supermarkets, most of the Indian supermarkets have everything you need, not just food. Try Big Bazaar, Hypercity. There may also be other supermarkets in Delhi now.
Cheap things in India - indian clothes like Shalwar, Saree, Suits, Leather shoes and belts, Suitcases, imitation jewellery (not gold), some food items like cheese, various creams that women use, books. Whenever I visit India, I buy a pair of branded shoes. They are expensive (around Rs. 10,000 in LKR) but they are very comfortable and durable.
Places to visit: Delhi is a ancient city full of history. You can easily spend one month to see whole of Delhi. The usual places are Red Fort, India Gate, Parliament House, Jantar Mantar, Kutub Minar, Humayun's Tomb, Lotus Temple (a Hindu, Muslim, Buddhist, Jainist multi-faith Temple), Jama Masjid, Nizamuddin's Tomb, Old Fort, Hauz Khas, Nehru Park. There are hundreds of other places in or near Delhi. Taj Mahal is the most popular. I think Taj Mahal is closed on Fridays.
Important:
Always wear a face mask. Delhi is one of the most polluted cities in the world. Particulate pollution in Delhi is extremely high
Avoid threewheelers as much as possible. They WILL cheat you. Some of them will even go to the extent and say there's a bomb explosion near your hotel and take you to another hotel. Try taxi booking apps like Ola or Meru. Best is to use the Metro.
Always drink bottled water from a supermarket or reputed shop. Dont buy water or drinks from roadside stalls. They have fake Pepsi and Coke in India.
Delhi climate is dry and hot. Wear a hat always and drink plenty of water.