You should be able to get a copy of all paperwork today if you ask. Then you have plenty of time to read things over if you wish to do so (or you can just blindly sign your name dozens of times and hope for the best). Don't worry, most of the paperwork is virtually harmless.
The biggest thing to look out for is the math. I've seen poor math myself on a house closing and I've heard about it from tons of people. It doesn't hurt to triple check the numbers. The most common problem that I've heard about is the taxes.
1) Are all of the back taxes truely paid? Title insurance should warn you of this, but you should double check that their math is correct. Suppose the property tax for 2009 is $4000. Since you are only going to live there for 105 days this year, you should only pay for 105 days of that $4000 and the previous owners should pay for 260 days. This math is frequently screwed up, and you'll be signing that you legally accept whatever math that they did. In this example, there are $4000 * 105 / 365 = $1151 of taxes that you owe.
2) Is there any way that there are taxes that you don't know about? Title insurance probably won't cover this. This problem stung my sister bad and another ATOT poster just mentioned the same problem in a thread yesterday. In my sister's case, she bought 1 floor of a 3 floor building. It formerly was an apartment with three one-bedroom apartments and she bought one of them. Since it was being split up just as she bought it, of course her new 1-bedroom apartment didn't owe any back taxes when the title company did a search. But the old 3-apartment building DID owe back taxes. When the goverment finally came around, they charged her for taxes of all three apartments for years that accumulated before she bought the place. A little legal action and she got that reduced to just 1 apartment, but still, she had to pay for the previous owner's neglect.
3) Do all of the fees actually match what were promissed earlier? Or did a fee suddenly appear or a known fee get extra padding? There will be dozens of gotcha fees (overnight mailing fees, realtor documentation fees from a supposedly free realtor, title insurance fees, etc.) Most will be legit, but at least make certain that they add up properly and match what you agreed to with the seller when you made an offer on the house.
4) Does the escrow money add up? You should pay for a full years of insurance plus enough money to cover the 2009 taxes (see #1). Then the monthly escrow payments will be to cover 2010 taxes and insurance (when they come due in 2010). In my case, they made me pay upfront for the year I bought it and for the next year then charged me a second time for the next year. True, they legally paid it back to me when they found the error. But they kept $3000 of my money interest free for a year. And no amount of pleading will get a bank to give you back your escrow account money any earlier even if they have that money by mistake.