Play the first mission (rescue a stalker from some raiders) and collect as many items as you can from the fallen. Go back to the camp you begin the game and sell the collected guns. At that point you should already be able to but the silenced handgun and have some spare change (ideally just over 500 RU), if not then do a side-mission for the stalker you just saved.

Once you have the silenced handgun and ~500 RU you should make your way north towards the bridge where you can see 4 or 5 army soldiers engaged with a few stalkers. Save your game and wait until the soldiers got rid off the stalkers and speak to the soldier towards the left of the bridge (close to the debris, not the one at the campfire!). He will let you pass the barricade for 500 RU. Agree to pay the money and he will turn around. At this point you have about 3-5 seconds where none of the soldiers will be looking at you, and the soldier you just spoke to just so happened to be in an area of debris and pipes. Shoot him with the silenced handgun, and MAKE SURE it is a headshot. He will go down quietly, however make sure you un-equip your weapon at the same time.

If you have done it quick enough the other soldiers will not notice a thing and you can loot the body to get hold of a decent weapon, grenades and some anti-radiation caps. However, timing is everything and since the solider's position changes during the fight you may want to make use of your quick-save/quick-load keys. Also, while you are at it, make sure you climb up to the wagon still hanging at the top of the bridge for some more hidden equipment.