I just did a balance transfer and am psyched how much money it is going to save, but I have a few tips for ya. In my earlier life I used to be a supervisor at a credit card bank so I have a pretty good bead on what to look out for (thus the long post heh)
As mentioned, verify there are no fees for the transfer.
Never, ever be late on a payment after the transfer or you might risk defaulting to a higher variable rate and lose the 0% intro rate and end up paying 20% or something. I think setting up an auto-pay out of your checking account and then religiously checking that the payment goes through properly each month is the way to go.
The order payments are applied to the balance is key. If you already have a pretty high balance at a higher rate on the 0% balance and then transfer onto it, the transfer isn't as good of a deal for you. That's because your payments each month will be going to pay off your promotional 0% balance first and not even touch the purchase balance you racked up that is charging a higher interest rate. But it sounds like this is a new card to you, so probably not an issue for you, but still good info. Know that credit card companies will always apply your payment to the portion of your balance with the lowest APR/interest rate first.
Make sure you don't transfer the full amount of your credit limit. You need to leave a cushion so you don't go over-limit when finance charges are added. Over-limit fees are often over thirty bucks these days, avoid them. Also don't transfer any charges you are disputing, like if a business over-charged your card for a rental or something. You will lose all dispute rights once you transfer the balance, because to the law you have paid the disputed balance.
On your other cards make sure you verify that the transfer payment hits, but still plan on making your normal payment for the month. It can take over 10 days for the transfer payment to arrive and it would suck to get a late on your old credit card account because you thought your balance transfer would take care of it.
Also on your other cards, even if you transfered their full current balance it will not pay off the card, i.e. you will still owe money due to the finance charges/APR that already accrued on the balance before you transfered it. So make sure you follow up the next month and pay off the cards completely after your account cycles. Alternatively if you are feeling brave you can call customer service and ask for a payoff balance and send the check today.
And make sure if either of your old cards charges an annual fee that you demand that they let you have the card for free or cancel the card because you don't want some annual fee to sneak past you and then cause a late pay history over a stupid annual fee.
Yet more on your old cards... on leaving them open or closed I would say close one of the two so that you end up with two major credit cards total (VISA, MC, AMEX, Discover). Two credit cards is enough to build credit, and having worked at a bank I can tell you that having lots of open credit cards can actually hurt too. If you have a 0 balance on all your cards, but say have something like $30k of available credit between all of your cards I might not be too excited about loaning your $30k for a car if you were borderline since I don't know if you will run out and max out your cards after you get the car loan. It's almost to a banker as if you owe the money even if you don't on your available credit.
But not to diss having some credit cards, say like two. You still need to build history, but keep in mind going crazy with lots of credit cards is a bad idea. Also, having an account open for a long time can have benefits on your credit as it makes you look stable and often the credit card company will score your account higher if its been opened longer so things like requests for a lower APR and higher credit limits are easier to get. Of course if the bank for your older credit card sucks then ditch it LOL.
Finally, don't get the idea that using endless balance transfers in a cycle of opening new credit card after new card is a good idea for long-term financial health. There are people who literally try to get a new card every 6 months endlessly to get a balance transfer deal and then ditch the account once the deal ends and open a new one somewhere else. Every time you apply for credit it registers on your credit score as a negative in a way. It's called an inquiry and basically when someone is constantly applying for credit it can tend to look desperate to a bank, especially if they have a lot of inquiries but don't make a lot of money. But even people who make more can be negatively impacted if they try to open too many credit cards when they try to refinance something down the road.
Hope this wasn't TMI, heh, but having seen people get burned I felt compelled to share. Best of luck, and congrats on at least having good enough credit to get the offer!