In the heart of the East Village you can still find affordable great places to eat. Gnocco is one of them! It offers Italian food with a lot of seating between the front room and an indoor garden in back. The pastas and pizza are delicious. The vibe is cozy and romantic, well suited to a date though you can also go with friends and family.
I recommend going family style so you can try the pizza, which tends to be large, and being receptive to staff suggestions. When you finish your meal, go next door to Blind Barber, a speakeasy whose front is a barbershop, if you’re in the mood for drinks.