The following is a list of weapons, their locations, and hints on how to obtain them:
Use a helicopter,boat, or swim to reach Happiness Island. Go inside the first door of the Statue Of Happiness to find a security guard with a carbine rifle. Kill him with your pistol and take the rifle.
To obtain the Desert Eagle before you can buy it, take it from the gun salesman in Broker after killing him. Note: Doing this will prevent you from making any future purchases at this location. This will also cause bodyguards with shotguns to pursue you.
Go to the Meadows Park area in Dukes and go straight for "The Monoglobe". Walk in the water to the center of the base and climb up to find hand grenades early in the game.
M-4 assault rifle
Go to the Easton area on Algonquin and at the Grand Easton Terminal, go up the stairs in front. Keep going around that floor until you reach a small pond. Move past it until you see a turn going left, then go to the left to get in between a tall building and a smaller one. At the end of the passageway is a dead end. Turn and you will see a glowing orange garbage can. Behind it is an M-4 assault rifle.
Swim to the island north of Algonquin and climb up the north side of the island. There, you will find a rocket launcher.
To get a good shotgun early in the game, go to the strip club. On the way out, there is a manager's office that you can enter that has a large Remington shotgun on the floor.
There are two smokestacks along the north coast of Algonquin. Climb the ladder on the eastern-most side to find a sniper rifle with a laser sight.