I am trying to attract the Merchant on my world and according to the wiki, I need at least 50 silver coins.
What is the fastest way to farm silver coins in a new world?
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Although not the fastest way to "farm" silver coins, as such - could you just invite a friend who already has 50 silver coins into your world as a multi-player environment, and then the merchant will spawn assuming you have the right house for him? A world made available on multi-player will still be available for you as a single player afterwards.
As Camster points out below, they would have to have 100 coins, and give 50 to you - as each player needs to have 50 coins before he will spawn.
If you're willing to risk going into the corruption, harvest two vile mushrooms. Turn those vile mushrooms into vile powder and sprinkle rabbits and goldfish with it. The resulting corrupt bunnies and goldfish are easy to kill one at a time and drop six and five silver coins respectively. After at most ten fights you'll have enough cash to attract the merchant.
I managed to get a lot of silver coins by killing the Jungle slime, they drop about 5 silver coins each time and they are simple to kill. Also don't die if your on normal mode as you will lose your precious silver coins...a good way is by using a chest and keeping all of your coins in there. You could also turn 100 bronze coins into 1 silver coin if you use a work table.
There are 4 basic ways to get money: Killing Enemies/Bosses, pots/chests, angler quest rewards and selling items to NPCs
Since the 1.3 update there has been an Expert Mode that will reward additional loot from chests, money dropped from enemies and better chance of rare drops. This will help get money early game.
Be Warned! While expert mode does have it's perks the enemies are much harder with an increased 2x Damage and 2x Health and those buggers can make off with your hard earned cash! If you can, locale the thief with their sparkly appearance meaning they have stolen coins of the colour they sparkle.
You also drop 75% of money (for software characters) so it is always a good idea to have a chest or piggy bank to store money.
Pots have a chance of spawning a coin portal that gives you 5-15 gold so another good strategy is to run across the surface and smash all the pots you can find.
The fastest way possible is to cheat. Download a warehouse map, get the money (and stuff if you want), done.
Warehouse maps such as this.