You can bring dollars with you (as it is very difficult to bring Indian currency from outside). Then, you can go to banks for exchanging foreign currency.
"should I take US Dollar or Indian Rupees?"
Make sure you obtain notes of 100 and 500 as after the demonetization, it is very difficult to get change for ₹ 2000 note
And how much will I need estimated per day?
It totally depends on where you will be going and how exactly you will spend your day. On a normal day with average shopping, and eating food outside, you could spend between INR 2000 to 5000. On a busy day, with lots of travel in Delhi, and visiting shopping malls, you might end up spending between 10,000-20,000.
In other words, what's the chance to pay dinner or taxis with credit card?
If you are in Delhi, you can pay with credit cards at food outlets (Domino's,Subway, KFC, etc.). But to buy anything from local vendors, thrift shops, and for travel, always keep change in small denominations.
Hope, this helps :)