This quest is completely optional. You can choose to end it at any time, but you will not receive the rewards Sunny gives you, and you will be unable to take the "By a Campfire on the Trail" quest.
After leaving the doctor's house, you'll have to speak with Sunny Smiles in the Prospector Saloon. Tell her you want to learn to survive in the wasteland and she'll take you out behind the Saloon and teach you to shoot with a varmint rifle. Shoot three of the Sunset Sarsaparilla bottles and Sunny will take you to hunt some Geckos. Just follow her to the creatures and then crouch down and try to get the jump on them. Once you kill them all, Sunny asks for your help in clearing some wells of the creatures. She leaves you alone to do this, so use your pip-boy and map HUD to find the targets. At the third well, you'll find a Goodsprings woman under attack. If you manage to save her, you will receive a reward of fame in Goodsprings and 3 purified waters.
This quest will lead to the quest "By a Campfire on the Trail," which is another optional quest.