The first explosion was in Baramulla district, 55 kilometres (35 miles) north of Indian Kashmir`s summer capital Srinagar, leaving a civilian and an Indian border guard dead.
A soldier and a civilian were also hurt in the blast, a police spokesman said, adding paramilitary border guards were the target.
In Srinagar, 13 civilians were hurt when a grenade hurled at a paramilitary foot patrol missed the target and hit pedestrians near the main bus terminal in the city.
The blasts sparked panic at both places and people ducked into shops and buildings fearing clashes between rebels and troops.
The blasts were the first major attacks by militants in civilian areas during the ongoing Muslim holy fasting month of Ramadan.
A separatist insurgency has raged in Indian Kashmir, a Muslim-majority state, for over 20 years.
The insurgency has left more than 47,000 people dead since 1989, according to official count. (M014/K004)