Responding 3...Nf6 to the Italian Game initiates the Two Knights Defense. This isn't an opening to play if you're looking for a calm positional game with material equality; in fact, it's been suggested that the name "defense" is inappropriate and the opening should've been called a "counterattack" or even "gambit" instead. White playing 4.Ng5 practically wins a pawn by force, and it's up to Black to demonstrate compensation.
After 4. Ng5 there are two lines: the Traxler variation with 4...Bc5 invites White to take on f7 with either piece. Black gains superior development as compensation. The resulting positions are very tactical (even by the Two Knight's standards) and complicated. Entering this line without preparation is asking for trouble, but at amateur level, you can outprepare your opponent especially if they naively go for 5. Nxf7.
The other line goes 4...d5 5. exd5 Na5 (5...Nxd5? is a "well-known bad move"; White can just play 6. d4 intending O-O and a straightforward attack) 6. Bb5+ c6 7. dxc6 bxc6. This is the same line given by D M in the other answer. White is up a pawn and has a much superior pawn structure, but Black is way ahead in development and has many moves that develop with tempo available. To illustrate Black's initiative, if White plays to grab more pawns with 8. Qf3, then after 8...Rb8 9. Bxc6+ Nxc6 10. Qxc6+ Bd7, the lead in development widens even further. Against the more conservative 8. Be2, Black still has 8...h6 9. Nf3 e4 10. Ne5, followed by 10...Bd6, Qd4 or Qc7 to drum up an initiative. Note however that after these moves, Black is still a pawn down with inferior pawn structure! If Black doesn't make use of his development advantage and attack, he will go down in flames.
1. e4 e5 2. Nf3 Nc6 3. Bc4 Nf6 4. Ng5 d5 5. exd5 Na5 6. Bb5+ c6 7. dxc6 bxc6 8. Be2 (8. Qf3 Rab8 9. Bxc6+ Nxc6 10. Qxc6+ Bd7) h6 9. Nf3 e4 10. Ne5
The upshot of all this is do not play the Two Knights if you're looking for a calm positional game with material equality. Play 4...Bc5 instead. The Giucco Piano can also become sharp and tactical if White goes for the Evans gambit (5. b4), but at least then you're the one with the material advantage.