I've answered this question a few times new in top 5's and top 10's, but my "must have" list- excluding bug fixes and stability mods -would include:
Roleplayer's Alternate Start - I refuse to play without it anymore, it allows me to build the character I want to play right off the bat with specialized skills, traits, and even a background. It also grants the ability to start at locations other than Goodsprings, sometimes putting you in NCR and Legion camps, or out in the middle of Fiend territory where you're immediately forced to stealth your way past hordes of enemies and get to safety, or blend in with the faction and join up with their cause. It's a fantastic mod that I would recommend to anyone.
https://www.nexusmods.com/newvegas/mods/45739
URWLNV Ultimate Edition v2.1 - This is the spiritual success to Nevada Skies, and a fantastically integrated weather system with various types of climates that can be easily adjusted to whatever you want the Mojave to be like. From scorching desert to snowy wasteland, you can have it all! It also removes that terrible, orange filter over everything.
https://www.moddb.com/mods/urwlnv-ultimate-edition-v21
More Perks/Traits - Adds the variety of perks and traits to the game that allow you to make a really versatile and unique character as you start and progress. These are defining character abilities, so having more options is a fantastic addition to the experience.
https://www.nexusmods.com/newvegas/mods/35979
Enhanced Camera - Probably falls more into the bug fix, stability, or utility category, but definitely one of my "must have" mods. I'm gonna include Weapon Animation Replacer here to.
https://www.nexusmods.com/newvegas/mods/55334 https://www.nexusmods.com/newvegas/mods/43240
Just Mods Assorted - If you're not using Project Nevada, use these instead. Trust me, you once you have them you'll never want to NOT have them again.
https://www.nexusmods.com/newvegas/mods/66666
A Familiar Friend - Pip-Boy 2500 - Just better to have the handheld than the wrist mount, for the purpose of fashion. You could also go with the Radius, but I like the retro style of this, and that it's a callback to Fallout 1 and 2.
https://www.nexusmods.com/newvegas/mods/38844
Advanced Recon Thermal Nightvision - Again, if you're not using Project Nevada, use this instead.
https://www.nexusmods.com/newvegas/mods/38328
Uncut Open Freeside - This actually makes that area run better for me, as well as being a better version of Freeside to begin with.
https://www.nexusmods.com/newvegas/mods/46355
Holster - Got sick of my handguns magnetizing to my leg.
https://www.nexusmods.com/newvegas/mods/34854
Ultimate NCR Overhaul - Just my personal favorite overhaul for the NCR, I think they look fantastic and still fit immensely well with the setting.
https://www.gunetwork.org/t10746-fnvrelease-ncr-ultimate-overhaul-v1-0-1Homemade PPSH - Not entirely a "must have" for me these days, but a really nice little weapon that fits in well with the jury-rigged, improvised, makeshift aesthetic of the post-apocalyptic setting.
https://www.nexusmods.com/newvegas/mods/50855
This is just a handful, and going beyond this would be tiny mods that are all about personal preference at this point. I think the majority of these are "must haves" for ANYONE thinking about picking up the game today.