To be a bit more specific than than the thread title, what type of:

A. Weapons (Primary, Sidearm, Reserve, Extra)
B. Armor(s)
C. Aid

...do you all carry when running around The Capital Wasteland or New Vegas? I realize many have more than one type of play-style as do I. I'm only listing my primary as it is what I use most.

Here is my primary load-out:

A. Weapons:

A1: Primary - M82 .50 cal Sniper Rifle with Silencer (Hotkey 3)
A2: Sidearm - either .45 Auto, 9mm or Jackal .50MG (Hotkey 1)
A3: Reserve - Double-barrel Shotgun (Hotkey 4)
A4a: Extra - 25mm Grenade Launcher (Hotkey
A4b: Extra - Dartgun (Hotkey 7)

B. Armor:

B1: Gopher Advanced Recon Stealth Armor, or
B2: Legacy of X13 - An Armor Overhaul

C. Aid:

C1: Stimpacks (Hotkey 5)
C2: Radaway (Hotkey 6)
C3: Anti-venom
C4: Venom (in the event I pick-up throw-able weapons to poison)

Tough, since it varies for certain characters I play as. I'll just do my current one.

A1- Some form of a rifle
A2- .44 Magnum
A3- A sniper rifle of some kind

B1- Brotherhood power armor

C1- Stimpacks
C2-Radaway
C3- Antivenom (for those pesky cazadores)

Silenced .45 auto, Assassin Suit and Silenced .50 GRA.  Ultimate one shot everything.  I'm usually packing Stimpaks and Radaway for aid.

That's vanilla. For modded, I use Dragbody's Splinter Cell Classic Op Set from the Sampler, a FN Five SeveN from the Nexus, and the M39 EMR that uses Millenia's textures. Then I use Stimpaks/Radaway.

My actual gear changes, but I keep roles associated with hotkeys...

1:  melee weapon
3:  general purpose firearm, something with ammo to throw away.
4:  heavy hitting firearm, bigger handcannon, when 3 won't do, switch to 4.
5:  automatic assault weapon, something I use for groups or feel like tossing lead for the heck of it.
6:  scoped firearm, whatever I'm using for ranged shots and not worried about ironsights being obscured.
7:  Some kind of explosive or thrown weapon.
8:  Whatever energy weapon looks fun to play with or I have too much ammo weighing me down and need to burn.
9:  Binoculars

Armor is also rather fluid and, considering mods, this becomes a question of style more than anything else. I prefer to customize or tweak something unique for my character.

For aid I just use whatever's at hand. Considering this includes food items I'll say Gecko Kabobs and Sunset Sarsaparilla.

Mine for NV would be

Weapons
Bushmaster Carbine
Anti material Rifle
Light Saber Pike/teal or black

Items
20 stim packs
5 doctors bags
5 Atomic cocktails
20 Purified waters
10 Steaks any verity
2 super stims

A Business suit/ I like the challenge, with hat

Food, water and med supplies ( love me some hardcore), pistol, smg/auto rifle, and .308 or higher. Usually try to have about 5 grenades ( or other throwable explosive) also.

My personal loadout is that for a stealthy/assassin type character, because that's what I like.

Weapons:
YCS/186 Gauss Rifle:  I like this as my sniper rifle, high damage and it gets the job done (I used to use an anti-material rifle, but the weight of the .50mg is murder.  I think just 100 of them was like about 36lbs.)
Marksman Carbine (All-American):  For when I've been spotted, the enemies are numerous but still not far enough away to warrant the use of my back up weapons, but also close enough YCS/186 wouldn't be very effective.
10mm Sub-Machinegun (Sleepy time variant from GRA):  Its silenced and is good for close to mid-range combat, especially inside interiors and with multiple enemies bearing down on you.
45 auto pistol (Light in Shining Darkness):  Good for close to mid-range combat in interiors or when there are just a few enemies.
Displacer Fist:  Good for anything up close and personal, plus I can use unarmed techniques like the ranger take down, also me like punch stuff.

Armor:
Light stealth suits such as the Asssassin armor from Dead Money, the t6m black widow outfit, or this 'casual' tactical jeans t6m outfit I made using nifskope.

Chems and items:
Stimpaks
Radaway
Rad-X
Purified/dirty water (if purified is unavailable) and food items.

I basically limit myself to only using modern/technologically advanced weaponry and equipment.  So no repeaters or revolvers, etc.  I also don't use any chems or alcohol.  So no jet, psycho, whiskey and so on, not even rushing water.

Well... If I'm storming any legion encampment, my loadout will consist of a minigun or grenade launcher, Enclave remnant power armour and a vast array of hand grenades/mines as well as any Fatman variant (just in case   ). In the worst case scenario I at least have a chance of taking as many of them with me. Otherwise I will normally run combat armour, a desert ranger helmet or headwrap, a marksman carbine, a 45 auto pistol, a machete, binoculars, stimpacks, radaway and of course more grenades. And I cannot forget about those authority glasses .

Right now in my new game i'm using almost all modded weapons and armor.

Hotkey 1- Colt M4A1 with extended mags and red dot sight

Spoiler:

Hokey 3- Makarov pistol with mods

Hotkey 4- L96A1

Armor- El Tigre Armor and Ballcap with glasses/1st Recon Beret.

Spoiler:

I'm getting some great ideas from the posts here too! Might start another character and copy your set ups for real.

While the weapons and gear used in my loadouts change constantly, the concept is always the same.

Weapons:
1. Non-scoped rifle, for taking on tough targets that are too close for a scope.
2. Shotgun. For general close range death dealing.
3. 2 Assault Rifles. Usually one is silenced. And they use two different ammo types. Necessary for putting out a lot of lead fairly fast.
4. Sniper Rifle. For the hard to reach spots.
5. Sleepytime. Always have this. Silenced and very powerful. One of my favorites.
6. Energy Weapon. Mostly because disintegrating someone is very fun from time to time.
7. Misc. Grenades for mass destruction.
8. Bowie Knife for taking on radroaches and other pesky bugs.

Armor:
1. Some type of light armor, usually from dragbody's pack. I usually bump up the DT to make it more viable, so I can look cool and avoid death.
2. Hat or scarf, cause it looks awesome.
3. Authority Glasses. No need for explanation.
4. Adv. Night Vision Goggles. I run my ENB very dark so its almost a necessity when working at night.

Aid and Chems:
1. Lots of stimpaks.
2. 2-5 doctors bags
3. Antivenom
4. Lots of rad-x and radaway
5. Enough chems to run my own pharmacy
6. 10 bottles of water
7. Handful of steaks and gecko kebabs.

It's a big loadout but I still only run at 70% carrying capacity. I'm likely to widdle it down soon though. And aside from Sleepytime and the energy weapon, all of my weapons are either from Millenia's weapons or from R.S.A.

I run a riot shotgun, Scoped hunting rifle, and a revolver. I wear the NCR combat ranger armor and a ranger brown hat. + Authority glasses , pretty simple loadout

well i run with

Weapons

1.Mosin nagant with a scope
2.10mm pistol retextured with a laser and a silencer

Armor
Couriers Fatigues from couriers cache
Revealing wasters scarf
Saboteur bag
Authority Glasses

That is pretty much it!