Flashback about leaving the vault.
You need to collect data from the monitors—it's not difficult, but to reach two of the five monitors, you'll need a grappling hook and a lever. The grappling hook is in the car (used on the ladder), and a fallen road sign can be used as a lever. Regardless of your stats, clearing the rubble will take three attempts. There won't be any other problems here. You can choose any dialogue option—it won't affect the rest of the game. If you tell the doctor you're afraid, you can skip the prologue.
Vault City (VC)
Erudition 50 allows you to access your true medical history on the computer near the dock.
You can request a stimpack or a first aid kit from the doctor (Speech 35+ or maximum health less than 35).
If you use Erudition on the computer near the Vault City exit, you can receive a stimpack (Science 25).
Stats are not important for convincing a citizen to leave the vault.
A cunning guard gives you the task of "following a couple." You just need to find them, talk to them, and choose your options.
Sergeant Stark gives you a quest regarding the Gerlach community (but don't rush to complete it prematurely).
If you talk to Harold, then the sergeant (or both), Harold will tell you about Moreau and the Hub. After this conversation, the doctor in the shelter will give you a quest to obtain mutated blood (pig rats are found in the HQ itself, coyotes are often found nearby, and you'll encounter plenty of geckos). For this, the doctor will teach you medicine recipes and tell you about infections. What he won't tell you is that infections in the game depend on your Naturalist skill—the higher it is, the less trouble you'll be.
Peter Graves gives story quests.
Sam returns your belongings (keep the crumpled paper!) and gives you the quest "Warehouse Special Assignment," which will take a very long time to complete. Also, if you steal his journal, you might get an interesting dialogue afterwards. During this encounter, you can gain additional experience by convincing him to touch you. The experience is awarded in two batches, and Charisma (higher than 2, but 6 is sufficient) is required to obtain the second batch. After this, you can blackmail him (-5 karma) to obtain the fuse. You can't obtain the fuse from both Stark and Sam.
Joshua can give you a quest to find the robot—no stats are required. The robot is located in the "Opal Mine" location, but to get to it, you'll need a stick of dynamite, which is found in the "Wasteland" location. The computer near it increases your Naturalist skill by 3% and unlocks the Nuclear Power Plant location.
Throwing darts at the target will grant you experience and a 5% boost to your Throwing Weapons skill if it's below 90%. You can also gain experience if you score more than 40 points.
At the Opal Mine location, you can find a holodisk (give it to Joshua to decipher), a ghoul in the dungeon—it's worth talking to him more closely (you need 9 Intelligence to fully complete his quest), a robot for Joshua's quest, and a pigrat for collecting blood.
At the entrance to this location, if you have at least 7 Luck, search the toilet there (this is not a joke! Even though it's funny). You'll get poisoned, but you'll find many useful items to get you started (including a stick of dynamite or a Molotov cocktail to break down the door).
Source: doc.