Walking through vault 88 and listening to begin again from dead money was honestly one of the most memorable things in gaming for me overall.

Was Seeing New vegas itself

Well i have two which are, one during when the Enclave destroy Liberty Prime and the other one is when you down The Prydwen.

Easily the first time I played a hardcore play through. I was on my way to red rocks with my #1 Boone and then all the sudden the message appears while fighting raiders "Boone has died" I cried to the heavens "HOW" and looked it up and found out my Boone could easily die in Hardcore mode I decided not to take any companions that playthrough

Shooting Burke in the head immediately after meeting him and hearing everyone say "Good riddance," "I could care less," "nice shot."

And then shooting Joe Cobb in the head after he bumped into me and said "What the hell is your problem?" Hearing Sunny shout "NOT MY PROBLEM!"

While playing as a Legion Spy, I had Boone as an companion at the time. Once he started to have his suspicions about my character I locked him in the freezer for him to be eaten by the guests at the Ultra-Luxe Casino.

Another memorable moment is destroying the Silver Rush by allowing a customer to come in with explosives. I stole every energy weapon in there with a Stealth Boy and using the grab key before hand.

It was a few years ago and I was playing through a quest mod. I had entered a interior infested with enemies. Only my "This Machine" in one hand and my GSh - 18 in the other. I had one of my corgis, Willow, Veronica, Niner, and Boone. We entered and instantly were under fire. I ran and found some cover behind a wall while Niner and Willow advanced. My corgi had ran into the thick of it and was going unconscious every few minutes. I had saw a door next to me and decided to go see if it would lead to a vantage point so I could flank them. I went through the winding corridors with Boone and Veronica at my side. Every now and then we encountered a enemy but I dispatched them with ease. We had eventually ended up behind the horde of raiders, I believe. And we just lit up on them. The four of us choked them out allowing no one to retreat. When it was over I secured the intel I was sent in to retrieve and took a long needed rest at my humble farm outside of Vegas. By far the most hectic moment in Fallout New Vegas.

For me there are several moments that have stood out. Definitely leaving the vault in FO3 was just awesome and mind blowing. I remember being amazed and awe struck at the visuals on my 360. Having replaid it several years ago while I was still impressed at what they had created it certainly has aged considerably. Also entering the strip for the first time in NV, I took my time getting there and so after hearing it constantly being referred to after about two weeks was awesome. In FO4 meeting Kellog was initially the biggest moment for me, which was then surpassed when I met father.

Wow...there's more than a few. Most of them from New Vegas, I always liked it more than FO3.
Anyway, from FO3 as others mentioned before me leaving Vault 101 was awesome. Especially after the that prologue, it had a huge impact.
From New Vegas, when I first entered Nipton and encountered Caesar's Legion. It was pretty sick. Heads on spikes, everything burning, crucified people and that creepy music. I hate them since...
So my second memorable moment is, killing Caesar for the first time. I felt so satisfied. And people loved me for it. Not only NCR, but pretty much everyone. Then I realized I have to fight Legion assassins every 20 minutes. Well....that's a nice bonus. I like killing them.
Two more is coming from mods. One of them: discovering Dragbody's "Heroes of the Apocalypse"
I will never be able to play a vanilla race again. Still waiting for a Schwarzenegger race tho' I'll replay the game immediately if I ever get my hands on it.
The other one is the "Willow" companion mod. It's just a masterpiece. Sometimes I felt like I play co-op with a living person. Especially through her "questline". She speaks much less afterwards.
And the whole "Old World Blues" DLC. It's so sick, creepy and...I don't know. The whole thing felt like a bad dream or something. It felt totally surreal.

Edit: One more I forgot. New Vegas: Being banned from playing at Gomorrah, bacause of winning too much caps. 10 Luck helps a lot. You just can't loose. Nothing feels better when you play Black Jack, got 19 and you confidently ask for another card.

From Fallout 3, entering Megaton for the first time. "Name's Lucas Simms, town sheriff. And mayor too, when the need arises." Amata kicking you out of the vault after you remove her father from power. Storming the Jefferson Memorial with the BoS.

New Vegas, to me, is mostly about the characters. Yes Man, Ulysses, The Think Tank, Veronica, Dean Domino, and Joshua Graham are the most memorable characters to me. And the Toaster. Can't forget the Toaster.

From FO4, seeing the Prydwen for the first time. The music swells, the airship dominates your view, and you're left with the impression that you should not mess with the BoS.

Story Spoiler for those who still care:

One of the most memorable Fallout moments had to be when I first tried FNV siding with the NCR and wanted me to kill Mr. House. I go over to his hotel get in, kill every robot in the building and then eventually opening Mr. House's life pod or whatever you want to call it. I was rushing through it because for some reason, House creeped me out. While the pod was opening, I spammed the interact button and just my luck, the animation bugs out and House's dilapidated body pops onto my screen. Needless to say, I shit myself.

They required me to go on a quest before they would let me in. Thinking it would be a short and easy quest, I agreed and set off for the place they called the Glow. The horror of atomic war was never so obvious to me until then.
- Vault Dweller's Memoirs

Being set on a fool's errand by the Brotherhood of Steel, realizing they had sent me to a death trap simply to get rid of me. Yet braving it with my meagre amount of radaway and rad-x. Moving through the burned husk of the West-Tek research facility while seeing my geiger counter ticking as the radiation slowly poisoned me. I retrieved the holodisc they wanted, I survived to make an exit (though badly poisoned).

So many games these days are horribly afraid of allowing the player to fail. In Fallout 1 it almost felt like you were supposed to fail, given how uncompromisingly difficult and deadly the quest was. But emerging triumphant felt like a real, tangible victory against all odds.

When I first entered into the Strip, I was amazed at the buildings and scenery. It also had one of my favorite characters, Yes Man