Let's get this out of the way: PvP gear is bad, mmkay?
Every time we talk about gear and gear strategies, we have to tackle the looming spectre of PvP gear. There are two sides to this discussion. The first side talks about feral druids and death knights who go into random groups and pull 7k DPS. (That's bad.) The other side points out that for some classes, in some slots, the raw stats (like, strength or agility) is so much higher on PvP gear that the gear is viable.
The crux of the issue is resilience. PvP gear has a stat called resilience that reduces the amount of damage you take from other players. That's indispensable in PvP, where burst damage will blow you up faster than you can sneeze. (Even with resilience, a rogue and mage will make short work of your health bar.)
On the other hand, it's relatively easy and fast to get some high-level PvP gear. Queue Battlegrounds and Tol Barad until you're blue in the face. Pick up season-current gear -- and voilà, good item level. (PvP gear is a little generous with item levels compared to PvE gear, because resilience is so expensive.)
So, yeah. You can shortcut into Raid Finder with PvP gear. But that's gaming the system and will tend to be a drag to the rest of your raid.
The new heroic dungeons
One of the quickest ways to get your item level higher is to do heroic dungeons. Sure, you get valor points (we'll talk about those in a second), but the big deal is that you can run them over and over. If you're in a rush to charge into the Raid Finder, kill Deathwing, and call the expansion over, then heroic dungeons are the way to go.
The new heroic dungeons drop item level 378 gear. You'll notice that's slightly higher than the 372 requirement; this gear won't make Raid Finder easy, irrelevant content, but it does mean you'll be perfectly capable of performing your task (well, at least as far as your gear is concerned).
This is the same item level of Firelands gear, though there were a few perks to actually raiding Firelands that resulted in slightly better stuff. At the end of the day, if your kit is made up of all 378 gear, then you'll be just fine in Dragon Soul.
Since this stuff is the equivalent of Firelands gear, the obvious questions becomes, "Should I just do Firelands?" Well, maybe for the experience and the joy of killing Ragnaros -- but that's all. A single raid lock or three won't do anything to make sure you're Raid-Finder-ready compared to an equivalent amount of time in heroic dungeons.
Valor points can make it happen
Valor points can get you to the magic number, but it's going to take a while. Current valor point gear carries an item level of 397. A few pieces of valor point gear will rocket you above the Raid Finder requirement very quickly.
Of course, there's a problem. You can only accrue 1,000 valor points each week. You're not going to see a new valor point piece every week, sometimes not even every other week. Relying on valor points to get you above 372 will take a while due to valor point gating.
Hit the Auction House
Patch 4.3 also brought new professions gear. Crafters can make these item level 397 pieces and sell them on the Auction House (or, you know, just wear them). Right now, the gear is incredibly expensive on the Auction House.
Take a look on your server's AH and see if you feel like the gear is reasonably priced. This is a personal decision. When you consider that heroic gear is perfectly viable for Raid Finder, spending the thousands and thousands of gold on crafter gear feels like a waste of money. (Time is money, friend.) Buying the gear is the best option if you have gobs of extra gold collecting dust or if you just don't have the time to run heroics.
Bottom line: Do heroics, buy if you can
The swiftest way to get into Raid Finder is to simply keep running those heroic dungeons. Give yourself a little boost when you've managed to collect enough valor points or if you have the cash to spend in the Auction House. Really, though, just run those new heroic dungeons. That's why they're there.
Visit the WoW Rookie Guide for links to everything you need to get started as a new player, from how to control your character and camera angles when you're just starting out, to learning how to tank, getting up to speed for heroics and even how to win Tol Barad.