Mine was "that gun" followed by "big boomer"

The Chopper, never used it though. Ever.

That Gun or Maria. I found That Gun, don't think I actually took it though (cause it was theft), so yeah, just Maria.

Mercy, actually.

I think my first unqiue weapon was Maria.

Ratslayer.

Mysterious Magnum.

That Gun then I used That Gun to kill the Lonesome Drifter to get the Mysterios Magnum.

Chance's Knife. I stupidly walked north from Goodsprings at level 1 and fought off a bunch of Cazadores. Died a couple times till I realized I could sneak up the mountain and drop dynamite on them.

My first unique item you say... would have to be either The Chopper or Maria (but idk if that should count, considering you get this during the main story)

Where abouts did you find that one?
I don't believe iv ever had it before.

It's carried by the Lonesome Drifter. He can be found on the highway just North of Novac (just north of the El Dorado Gas Station to be exact) and he is sitting down next to the Sunset Sarsaparilla Billboard.

If you complete the quest Talent Pool, you can recruit him to sing at The Tops, and if you pass a skill check (either barter or speech; I can't remember) he will give you the Mysterious Magnum. I just killed him in my first playthrough so that I could get the magnum early on.

The magnum itself is a reference to the Mysterious Stranger's .44 magnum, and it has some...special properties. It is hinted at that the Lonesome Drifter's father was the Mysterious Stranger, and that the Stranger passed his gun down to the Drifter.

Damn, can I still get it even after I complete that quest? I'm fairly positive I did already.

Side question, is there really a show at the Tops? I've waited from 8pm till 12 but nothing ever happens.

Ratslayer.

After you complete it? I don't think so. Not without killing or pickpocketing the Drifter. You can always get him to unequip his magnum by reverse-pickpocketing a more power weapon onto him, and then pickpocketing the magnum off of him.

I think there is a rad pack performance at some point, I don't know exactly when though. It's before 8pm though, try waiting starting at 5pm and you should catch it. After Talent Pool is completed, all of the acts perform at certain times.

Edit:http://fallout.wikia.com/wiki/The_Aces

Ooh, good idea. Il try the reverse pickpocket idea.
Thanks m8.

Chopper, but it wasn't great. Then That Gun, same thing with chopper.

Rather fittingly the first unique weapon I found was Lucky which is my favourite weapon in the game.

The most important skill for me to raise quickly in Fallout 3 was Lockpick so I thought I'd do the same thing for New Vegas. I had a Locksmith Readers mag, but I didn't have enough skill with it to open that Hard locked safe. I went off towards the Mojave Outpost, did a few quests there and I don't think I was in Nipton yet when I got enough with Comprehension. I fast-traveled back to the Bison Steve because it was the first Hard locked thing I had seen yet in New Vegas and I needed some caps. I opened the safe, found Lucky and the rest is history.

I think it was The Ratslayer. Got it early on so I remember using it a lot.

That Gun, simply pick a easy lock then BAM! You have a futuristic magnum that fires AR rounds
I think Oh Baby! was my second, or one of the power fists, Pushy?

Maybe love and hate or lucky

Lucky. Investing in your lockpick early was a lesson I've learned in FO3 already, and it quickly paid out in FNV, too.
With that, and the not too far away located Ratslayer you're pretty darn well equipped in the early game.

that gun I believe, either that or the CZ67 avenger.

Ratslayer......followed closley by Lucky. It would be many a while before I would make my next find with the Q-35 matter modulator and really start searching for more of the uniques.