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A free shovel can be found at the second watering hole at Goodsprings source, going there right after leaving Mitchell's house does not take the player out of the starting area and there are no geckos.
Locate all the graves at the cemetery and start digging for some nice loot.
Detailed walkthrough
The goal is pretty simple. The player must buy or find a shovel and start digging up some graves at the Goodsprings Cemetery for loot inside. Unlike in Fallout 2, you won't be called a "Grave Digger" and your Reputation is not affected either.
There is loot to be found in the graveyard, namely a snowglobe which can be handed in to Mr. House for 2000 caps as well as a couple of cigarette butts laying around the player's grave which will serve as evidence in incriminating Benny. The grave loot is randomly generated based on level, therefore the player is not guaranteed to find anything of significant value other than the snowglobe itself.