Cheapest Hotels are available near New Delhi Railway Station - Pahar Ganj Side, at Pahar Ganj you can find more than thousands of Hotels of all range; these days one can get a Single Room with attached bath in just INR 500-600 and can get at lower cost if you visit in the narrow street like Chandi wali Gali etc. If you want to visit Delhi with family then I recommend you to visit Hotel Balaji Dx or similar in Paharganj; and the room rate of double occupancy is approximate Rs. 1800/- including Buffet Breakfast. Also one must have the power of bargain to get the cheapest hotel in Paharganj. Please note that never given attention to the person while coming out from Railway Station as they will get a commission from Hotel and this will cost to you. Also, you can find a hotel with price INR 400/- and always take care of your belonging in the cheapest hotels or always carry the same with you, if you want to leave the room during the stay. I always prefer to stay the cheapest hotel in Paharganj while travelling solo without any costly items.
Regarding Taxi service, you can get service of prepaid taxi service at New Delhi Railway Station with nominal charges. Also, you can avail Radio Taxi Cab service i.e. Mega Cabs, Taxi for Sure etc. and can be booked online and they can pick you from the station when your train reach. Now I always use Metro service as this is the cheapest way and for metro, you should need to visit Ajmeri Gate side of New Delhi Railway Station instead of Paharganj Side. After reaching your nearest metro point of your destination you can avail services of Auto Rickshaw to reach the destination.