Each weapon (including grenades and the whip) has 2 upgrades that count as additional weapons. The upgrades carry over even if you start a new game on another difficulty (on the same save spot).

Weapons	How to unlock
2nd level Assault Rifle	Box in Mission 5
2nd level Grenades	Clear Missions 1, 2 and 3 in Normal difficulty (S rank)
2nd level Rocket Launcher	Clear Missions 9, 10 and 11 in Normal difficulty (S rank)
2nd level Shotgun	Box in Mission 11
2nd level SMG	Box in Mission 3
2nd level Whip	Clear Missions 5, 6 and 7 in Normal difficulty (S rank)
3rd level Assault Rifle	Beat the game in Ultra difficulty
3rd level Grenades	Clear Missions from 1 to 8 in Hard difficulty (S rank)
3rd level Rocket Launcher	Clear Missions from 1 to 12 in Ultra difficulty (S rank)
3rd level Shotgun	Beat the game in Expert difficulty
3rd level SMG	Beat the game in Hard difficulty
3rd level Whip	Clear Missions from 1 to 8 in Expert difficulty (S rank)