As for non-vegetarian dishes, i recommend Mutton Biryani ( popular lamb rice dish with saffron and other key Indian spices), Tandoori Chicken (grilled chicken with Indian masala), Seekh Kebabs ( minced lamb patties), Murgh Makhani ( chicken marinated in yogurt cooked in gravy filled with Indian ingredients and garnished with a dash of butter/fresh cream) , Malabar Prawn Curry ( Coconut based curry with prawns) and Goan Fish Fry ( pan-fried fish fillets marinated with Indian spices).
In case of vegetarian dishes, i recommend Palak Paneer ( popular & tasty Spinach curry with cottage cheese), Naan ( flattened bread either oven-baked or cooked on pan), Tandoori Paneer Tikka ( chunks of cottage cheese marinated with Indian masala and grilled along with other fresh vegetables), Masala Dosa ( extremely popular Rice pancake stuffed with potatoes and served with accompaniments of sambar ( dal stew) and 2-3 chutneys ( coarse mixture of coconut with corinder, tomato or onion) and Kathi Rolls ( Indian Version of American frankie- flattened bread filled with vegetable stuffings and Indian spices).
Note: Paneer or Cottage cheese is lacto-vegetarian food.
For More information, Please visit Wikipedia and type Indian Cuisine- it provides you a comprehensive knowledge of Indian dishes.
Also Please Check out India's Superchef Sanjeev Kapoor's dishes ( both veg and non-veg) on Youtube
and India's reputed food writer Tarla Dalal's healthy and tasty vegetarian dishes on Youtube.
Kindly Note that India emphasizes and specializes more in Chicken, lamb and seafood dishes. For more details in Indian beef and pork dishes, type Goan Cuisine on Wikipedia.