Speech, because I love speech, it helps me a lot.

Guns - they're all over and everywhere. It would be unwise to go without them.

Sneak first, Guns second and third - repair. Because I mainly use fully upgraded hunting rifle and play as sniper, with hight Perception and average gun skill and related perks, sneak critical damage is extremely deadly and provides quite a punch even when killing deathclaw alpha male, deathclaw mother and legendary deathclaw.

Speech, because I am nothing without a silver tongue

I tend to go for Guns and Repair first. My weapons usually degrade and I can repair my weapons on the go while I find weapons on looted bodies.

On one of my playthroughs of new Vegas I maxed out Sneak and put Guns up to about 70. The game is pretty easy when you can sneak up on enemies and decapitate them with the infiltrator.  

Always speech, I love to speech myself out of situations!
After speech guns so I can use some guns proper without shaking a lot.

always my main weapon first.

Always speech first then lockpick

Every play through I usually try to max out my guns skill and then bring up my speech skill.

Speech

Depends on what type of play through it is, but for most of them i would normally max out guns speech and sneak first

Usually the essential skills such as lock pick, science and speech are the first skills I max out. Depending on what kind of play through it was of course.

For me it Survival Guns and Repair. It's a must in hardcore mode

Generally speech. But I find it a double edged sword.

Sure, plenty of places have enemies that are aggressive by default, but that's what guns and companions are for, but the majority of important encounters can be resolved none-violently.

A character with 100% speech basically becomes the Doctor from Doctor Who. There are few important situations that can't be solved by the player-characters fast and persuasive talking. And this is awesome. But... it can render tense situations not so tense, as you know full well you will be able to talk your way out. It also tends to render the final scenes of the games (The Master/The Enclave Squad/Autumn/Lanius and the General) toothless feel-good fests. A build up, to a fizzle.

I generally limit myself to around 95% Speech. This way I can have my character be practically the slickest sucker in the West (or East), but the biggest tests will generally be failed, or in FO3 terms there is still a chance to fail. It stops the skill from pretty much removing all tension and suspense, and actually gives combat skills a use in important situations, or alternate checks that would be otherwise ignored in favor of speech.

speech tots! makes stuff soo much easyer

I usually max out lock picking and science first I find those two skills the most useful

Speech is always the first one, than science and lockpick.

Lockpick and science. I hate coming to a lock door or safe that i can't get through. Sometimes, at lower levels, I'll come across a door that requires one of those skills but my other skill is the one I just got to the max. I get angry and just disable the door and poof. It no longer exists. Just like it should. Sometimes I do that to the criers on the strip. Killing isn't just good enough. Sometimes making them nonexistent is better (maniacal laughing).

Most of the time guns.

I hate speech in new vegas because it is extremely overpowered. For example it lets you skip whole queststages!
Most of you wrote that you max it out first nevertheless. Try out a character with speech below 20. That makes the game a much bigger challenge.