Fallout 1 allows a determined player to gain very powerful armor and weapons early on, via the Brotherhood of Steel, after which the game becomes ridiculously easy.
Or you can simply go back to Vault 13 after leaving the map-square with it and re-enter. Double-barreled shotgun, additional stim-packs and medkit don't sound that powerful, but they will allow you to waltz through everything till Hub, providing enough fire-power to drop dead any rider or mutated scum on your way there.
On a lesser scale, you can skip Shady Sands, go to Junktown, provoke the doctor into attacking you, and get his bodyguards' shotgun and leather armor - all nice starting bonuses.
In Fallout 2, there was a much-publicised, non-cheating method of obtaining one of the game's most powerful suits of armor before even reaching the second town of the game. It involved running nigh-naked through towards what amounted to the last town, and praying an Enclave patrol didn't show up as a random encounter and slaughter you messily.
And then you go to the NCR, where the first merchant you encounter is very easy to steal from. This allows you to get several pretty good weapons along with the Bozar, one of the game's most powerful weapons, and tons of ammo for everything.
Fallout 3 offers incredible chances at getting powerful equipment early, especially of you're willing to explore the world before embarking on the main quests.
The first two games have a steady armor hierarchy: Leather Armor -> Metal Armor -> Combat Armor -> Power Armor. The exception being the Power Armor, which you can get much earlier than you would expect. Not only will you only be using the Combat Armor for a short period of time prior to getting the Power Armor, but you might even find it beforehand—perhaps even before the Metal Armor.
Fallout 3 is a different story. The first shop you'll come across (Craterside Supply) offers a steady stream of Leather Armor, however at random intervals they'll have a set of Combat Armor available. The massive Damage Resistance spike makes the early game quests a cakewalk, and you'll be able to repair it by waiting for it to reappear, or by going to Rivet City which sells it in bulk. Expect to use it for pretty much the majority of the game, even when you gain the ability to use Power Armor since it has nearly the same DR but weighs significantly less.
For the melee player in Fallout 3, there's a readily available equivalent to the Disc One Nuke which will see you through the entire game. The Shishkebab, the game's most powerful melee weapon, is crafted from items laying in close proximity of each other in your warm up dungeon (Springvale School), and a schematic obtained from a merchant in easy reach of your first base town. The price may be high, but far from unobtainable at that level of the game. Although the final upgrade to the weapon's schematic comes late in the game, even at suboptimal power it's still the superior weapon.
Averted when it comes to powered armor, though. You can get it at any time — but to use it, you need training, which is only available after a story event.
If you have Operation Anchorage installed, you can complete it straight out of the Vault and are awarded Power Armor Training, as well as a suit of T-51b Power Armor that never degrades (and avoids the Agility penalty of T45-d and Enclave power armors). Alternatively, if you don't like power armor, there is also Chinese Stealth Armor in there, and among the stashed weapons is a Gauss Rifle and Jingwei's Shocksword. Oh, and please put the armor on immediately as Defender Sibleyand his mooks will turn on you shortly after.
Most minmaxers will leave Vault 101 with 9 INT and then beeline straight to Rivet City for the Bobblehead before reaching Level 3 in order to obtain the maximum amount of skill points possible. They also max out Big Guns without ever putting a point into it by abusing the respawning Flamethrower Recipies book in Bethesda Ruins.
Operation Anchorage itself is a Disk One Nuke — you get the game's best armor, the game's 2nd best ranged weapon, and one of the best melee weapons all from completing the same mission. Mothership Zeta on the other hand, awards you with insanely powerful weapons, ammo, and the ever-awesome Alien Epoxy, that you can use to fix any weapon, including rare ones like Tesla Cannons and Gauss Rifles. Also, the EX-B Drone Cannon is a freaking grenade launcher, the only one of its kind.
You can complete a quest in Rivet City early on and get a unique Plasma Rifle, with a higher ammo capacity and damage output.
Lincoln's Repeater can be found in a museum, but can be easy to miss if you are not reading a guide.
The Xuanlong Assualt Rifle is a unique Chinese Assault Rifle with boosted damage and an increased clip size. It can destroy most low-level enemies and remains useful throughout the game. All it takes is completing a somewhat Guide Dang It unmarked quest to spawn it, which is performed at an early location in the main quest line.
With the DLC Broken Steel installed the companion Dogmeat which without the DLC has health around 500, levels with you and starts at 2,500 hp at level 1. Later hits around 15,000 when you hit level 30. Since he's coded to be a higher priority target than the player's character, Dogmeat will draw attackers' attention away from the player character, meaning that enemies will attack a near invincible companion leaving you to attack freely without having them attack you. Dogmeat can be found as soon as the vault is left for the first time.
The "Alien Blaster". Although the gun is very powerful, its ammo is very limited and it's impossible to obtain more (since it's of extra-terrestrial origin).
There are more alien weapons if you download the "Mothership Zeta" expansion. And more power cells, too.
The first set of schematics for the Railway Rifle can be found in a power station just north of Minefield, where you go while researching for the first chapter of the Wasteland Survival Guide. Its parts can be found in the town's houses.
Fallout: New Vegas has, among others, the well-known Ratslayer, a unique Varmint Rifle with all the mods, found in a cave a little ways off the beaten path. The problem is you have to fight the bigger Giant Rats to get it, but it is well worth it if you can.
For Survival/Hardcore Mode players, the most powerful healing item in the game, Desert Salad, is amazingly easy to craft; even the relatively rare Pinyon Nuts (as compared to the absurdly easy to find Barrel Cactus Fruit and Brahmin Steak) are sold by most vendors, and there are a couple of places where they're guaranteed to be found very early in the game. At a minimum of 14 HP restored per second by the time you have the 55 Survival required to craft it, it makes the more difficult enemies cited in other examples of this trope much easier to weather.
At the start of the game, if you go east from Goodsprings to the Yangtze Memorial, you can use a shovel to dig up a grave, which usually has some form of mid-level weapon in it, ranging from Plasma Rifles to Incinerators.
North of Nipton, you can find a dead Bright Brotherhood ghoul. If you're lucky, you can get a Plasma Defender, a small, light energy pistol that's faster and almost as powerful as a plasma rifle while using ammo that costs half as much.
If you can evade the Cazadores north of Goodsprings, you can find a grave with Chance's Knife, an excellent melee weapon.
It's even easier to pick up Knock-Knock, arguably the best melee weapon in the unmodded game, and you can do it as early as Level 2 if you know what you're doing (and have both Stealth Boys and anti-radiation meds).
Cross Boone with ED-E, the first companion you can hire, and nothing will touch you.
Obtaining 50 bottles of Sunset Sarsaparilla, Save Scumming after drinking one until you get a Blue Star Cap, and repeating until they are all star caps gets you Pew Pew, a laser pistol that hits like a shotgun and does about four times the regular laser pistol's damage. If you have high Energy Weapons, this gun will carry you until the end of the game.
Oh, and you can steal the various weapons in the Van Graff store by dragging it to the toilet and then pocket them there. Among them is a Plasma Caster, a Gattling Laser, and a Plasma Defender.
Smalltime. With judicious tailoring of your character, use of stuff given or found in the first town and the right perks to max out your Lockpick, you can obtain the Gobi Campaign Rifle at level 4. 'Course, you'll be at least level 7 before you can fire it well.
Lonesome Road added Satchel charges, a placed explosive as damaging as the extremely rare and expensive C-4 charges; strong enough to take off the majority of even a max-level Courier's health. Though only moderately useful in the DLC itself, just having Lonesome Road installed allows craft them in the main game immediately. It requires no schematic or perk, just 25 Explosives skill and a few parts you can find lying around everywhere (lead and pistol powder you can find from the most common ammo, scrap electronics, and a sensor module) and you have something that can kill most things up to mid-game, often several times over.
Although in poor condition, and you have to fight a number of Fire Geckos to reach it, a Fat Man can be found in a crater to the southeast of Searchlight, and (unless you have the Wild Wasteland trait) three Mini-Nukes are present in the eastern church basement in Searchlight itself.
The Reinforced Combat Armor can be found in Walking Box Cavern, just off the road to Novac. The only trouble is that the cavern is inhabited by a pack of Nightstalkers, which can quickly do in a low-level player.
Ditto Fallout Tactics, it is possible to get some weapons early. You just need some poisons or drugs. You can try to get the brotherhood bunker encounter and kill the 2 guards using poison or drug overdose and get their miniguns, or poison the merchant and his guards to get Pancor Jackhammers and the Browning machine gun (though good chance you are not strong enough to use it yet, and ammo is rather scarce early game). If you get one of the merchant encounter, you can get EMP shotgun shells as well which let you easily bust the turrets in Preoria mission, along with a FN FAL rifle. If you get the Radscorpions fighting civilians encounter early enough and let the civilians die (which is easy, because they're fighting the Radscorpions with their bare hands) and loot the corpses, you'll find out that at least one of the civilians was carrying a Combat Shotgun.
You can also use the gambling bug. Bet nothing for the merchant's entire inventory, click on gamble repeatedly, and after 1-2 minutes, you'll suddenly have his entire inventory...