I have heard that shooting the first arrow to start the battle is very important to every battle.
Who shot the first arrow to start the battles of Ramayana and Mahabharata respectively?
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Let me address each of these two battles seperately:
Ramayana: The first fighting in the battle of Lanka started shortly after Rama and the Vanaras arrived at the gates of the city. Sugriva saw Ravana, and he immediately he challenged him to fight, as described in this chapter of the Yuddha Kanda of the Ramayana:
On seeing Ravana, Sugreeva got up all at once, while Rama and the chiefs of the army of monkeys were looking on. Gathering up his strength and courage, Sugreeva, in an impulse of fury, bounded from the summit of the mountain and then sprang to the top of the gate (where Ravana was).... The two warriors, Sugreeva and Ravana, who were skilled in wrestling, executed innumerable and myriad evolutions, taking up diverse postures, moving in a curved line like an ox's urine, coming and going, stepping side ways, having a retrograde motion to avoid blows, turning about abandoning the attack, dashing towards each other, leaping, standing firm and erect, retreated, turned sideways, rushed in a bent posture, ran lifting their foot to kick the opponent and by letting go or stealing away.
So this was just a wrestling match, with no weapons involved. The first time an arrow being shot is explicitly mentioned is when the Vanara Sampati fights the Rakshasa Prajangha, as described in a later chapter of the Yuddha Kanda of the Ramayana:
Sampati, who was struck by Prajangha with three arrows, killed Prajangha by an Ashvakarna tree, at the zenith of the combat.
Mahabharata: The fighting in the Battle of Kurukshetra commenced when Duryodhana and his brothers started launching arrows at Bhima as he was approaching them, as described in this chapter of the Bhishma Parva of the Mahabharata:
And roaring like a deep mass of clouds, and assuming an awful form, that hero [Bhima] frightened thy sons and fell upon them. Thereupon the brothers, viz., thy sons Duryodhana, and Durmukha and Dussaha, and that mighty car-warrior Dussasana, and Durmarshana, O king, and Vivingsati, and Chitrasena, and the great car-warrior Vikarna and also Purumitra, and Jaya, and Bhoja, and the valorous son of Somadatta, shaking their splendid bows like masses of clouds exhibiting the lightning's flashes, and taking out (of their quivers) long arrows resembling snakes that have just cast off their sloughs, surrounded that mighty bowman rushing (towards them) covering him with flights of arrows like the clouds shrouding the sun.
But it doesn't say which of the 100 Kauravas shot the first arrow.
It is interesting to note, by the way, that in both cases, the bad guys were the ones who shot the first arrow.