I suck at New Vegas. On my first playthrough I sided with Caesar's Legion (which was fine, I'm usually the bad guy in these types of games) and blazed through his missions, not really bothering with side stuff (I was leaving it for the end) and soon enough the "Point of No Return" came up and I began chipping away at side quests I had logged during my adventures. One of these was the Wild Card quest line. I unwittingly undid everything I had worked for with Legion, but saw the quest to the end, deciding to start again on another playthrough and do things correctly this time. 
 
It has occurred to me that I have no idea what I'm doing with the leveling aspects of my character. I know what I want but I don't know the best way to get there. I want to be the "Smooth Talker" that uses speech and bartering to get what he wants rather than violence. I try and focus on skills like speech, barter, and lockpick, but it's difficult to do while at the same time making sure my other skills aren't completely screwed. I find that I'm constantly out of stimpacks, ammo, and most importantly, effective weapons. 
 
I realize these things take time to build up, but how can I build up my character and explore the world when there's a Cazador or a Giant Radscorpion lurking just around the corner with enough poison to make mincemeat out of me in just three hits (or in some cases, less!)? I try and use VATS whenever possible, but it would be nice if my weapons did some actual damage on my opponents. The worst and most embarrassing part of it all is that I'm already playing on very easy difficulty. 
 
TL;DR: How should I level my character and what should I focus on so that I can survive getting from point A to point B without getting obliterated in the process?
 
EDIT: I don't plan on siding with the Legion this playthrough, that was what I did my first time, so don't let that get stop you from providing insight.
EDIT 2: It should also be noted that I'm still somewhat new to RPG's, which can explain in part why I am clueless with this stuff. Aside from Fallout 3 and Mass Effect, I barely have any experience with these types of games. I'm not making excuses, this is just for the "Very easy? How???" responses.

Very easy?Man.
When I played, I searched everything that wasn't guarded by huge mutated monstrosities.You never know when you'll find a good gun or a stimpack.Also, my problem was that I couldn't pick locks, and since you are focused on that, start looting.Also, If you're going evil, might as well kill people and steal their shit.I know I sometimes did.

I levelled up Speech, Guns, Science, Lockpick, and maybe sneak, all to 100 and then repair and medicine to somewhere high. I ignored the rest. I became pretty much immortal and had more than enough of everything. So maybe that's a good number of skills to focus on?

You've got to try and max out guns as soon as possible, and get the perks that increase the amount of ammo and caps you get when looting. Make sure you take whatever ammo you find, regardless of whether you use it or not. Just sell the ammo for guns you don't need. Also, having a companion doesn't hurt either (I rolled with Boone for most of the game, but he shoots Legion folk on sight, so that might not jive with your play style). 

Start with a very high intelligence. Get educated as fast as you can, then get comprehension (Don't get any perks that affect xp). Tag speech ,science and lockpick. Use Guns as your primary combat skill. Focus on leveling up Guns, Science, Lockpick, Speech, Repair, Sneak. Search every container you come across.

The most important thing you can learn is what to pick up to sell. Cigarette cartons and packs are worthwhile to pick up, but anything that weighs more than a pound or so should be ignored unless it's armor or guns. Always loot corpses, and always repair your stuff if you have lots of duplicates. Sell to Crimson Caravan, Gun Runners, or the Med Clinic. If you keep doing this on a regular basis, you should be rolling in caps and therefore be able to buy anything you need, including stimpacks, better guns, and ammo.  
Also, even if you want to talk your way out of things, you're still gonna need to fight stuff on a regular basis. Make sure you have high skill in either Guns or Energy Weapons, and stick to one until it's at 100. Gun Runners has tons of awesome (but expensive) Gun-type weapons, while the Van Graffs (Silver Rush Casino in Freeside) have some good Energy Weapons.

 If your fighting Caz or Giant anything, your right.  Your doing it wrong.  Your in the wrong areas and you should leave those places alone till around level 10+.
 
(If your stuck, just un-equip your weapon and run.  You can out run anything in the game.  Just move left to right to dodge leaping attacks.  This included deathclaws)
 
Never max out any stat.  There are implants, mags, books and other ways to finish the job. Its a waste that early on.  Most of those points dont even register to your stats.  Google it, they literally do nothing early on.  For your first levels, all points go in offense.  Guns, melee, whatever weapon you plan on using.  Do all optional side quests.  Raid Doc Mitchell's house at the start, he is your father and everything there is yours/green.  VATS isn't for killing everything, actually aiming will use less shells and cause more damage.  
 
Never use VATS in combat.  Its a waste.  Its only in-game for fun as a relic mechanic. 

 After you finish in good springs, head to primm, and so on.  Follow the main story line till you have enough supplies and power to get by.  Once you meet the legion on the way to Novac, try and go for Veronica at 188 Trading post.  She is powerful and only requires that you talk to her once.   No quests and you get great protection. 
 
After that your pretty much all set.  Once you get Ver. you can just explore and create what ever story you want.   With her you can also hit up the Brotherhood for quests and gear.  That whole place is a gold mine, if you want to raid it.  Single kills earn you a few thousand caps each.   Their armor and weapons sell for thousands.  Just run it over to the Gun Runners.  The robot has tons caps to trade.

Stop just following quest arrows. Wild Card did everything it could to tell you that it wasn't the route you wanted to follow if you wanted to support the Legion.

Like several people have said before me, a high intelligence is an absolute must if you're having trouble with your skill count. I don't know how hardcore you are but if you want to maximize your skills  I'd advise setting your intelligence at 9 in Goodsprings and then making your way to the New Vegas Medical Clinic and buying the intelligence implant (after winning 4,000 at the atomic wrangler) before you grow a single level.  Also, get the educated perk as soon as possible and then get the comprehension perk and use online resources to find skill books. Once your lockpick and science and speech skills reach 80 there is really is no need to increase them any more because you can temporarily increase them to 100 with magazines if you absolutely need to. Once you get your speech to 50ish you really don't need to increase it anymore until you get near the strip since, in my experience, that's where all the high speech checks start happening. I'd also advise setting your endurance to at least 8 at the beginning so you can get all but one of the implants.  As people have said before a good companion and some points spent in your primary combat skill should keep you alive for the most part and once you get to the endgame you should be at a high enough level that you shouldn't need to worry anymore.  Finally, you should really do as many side missions as you can, they grant experience, caps, sometimes companions, and of course, loot.

Don't ever forget the usefulness of sneaking, if you are "hidden" you can steal very useful things from people  in basically broad daylight, and the 'caution" or "danger" notifications are extremely useful, especially when you are still underpowered, as you will most likely be if concentrating on non combat skills..  My character is like level 16, I have speech at like 80ish, guns at like 50- 55, energy wpns about the same as guns, and I think my repair, sneak science and medicine are all somewherearound 45-60 or so, I think lockpick is like75.. 
 
I'm playing in "hardcore" mode on normal diff and I'm constantly dealing with my weight limit as well as hunger and dehydration.. 
 
I'll tell you that using sneak effectively is a life saver, granted it takes a lot longer to play the game this way, I'm over 60 hrs in and like I said only lvl 16, I just finished helping the Boomers at Nellis, haven't done any wild card stuff or ceasar questline yet.. Tons of exploring tho.. 

I feel like a retard also, I totally just remembered that I still haven't started using a house yet.  In FO3 my house was stuffed to the gills, and now when I could really use it, I completely forget about finding a house to base my operations and store equipment...  Sixty hours in too!

You cannot spread your points out too thin. Focus on four skills. To make things easier you should definitely get you guns skill up. 
 
Also, USE MORE SKILL.

" Start with a very high intelligence. Get educated as fast as you can, then get comprehension (Don't get any perks that affect xp). Tag speech ,science and lockpick. Use Guns as your primary combat skill. Focus on leveling up Guns, Science, Lockpick, Speech, Repair, Sneak. Search every container you come across. "

Thank you all for your input. I've edited the OP to avoid further possible confusion.

Run backwards while keeping your trigger pressed in. That's how I play Fallout.