There's no way that you, as buyers, can expedite the process.
You make an offer on a short sale. The seller accepts or rejects it. (There isn't too much incentive for the seller to reject it, since--regardless of the price--the seller ends up with nothing. The only incentive is in accepting an offer that the bank is likely to accept.)
Once that's done, you enter a black hole. The listing agent provides information to the lender(s)--the hardship letter, the seller's tax returns, W2 statements, and so on. And the listing agent stays in regular contact with the lender. You, as the buyer, are totally out of the picture.
The process may go quickly--a month or two. It may take a year or more.
Again, you have no control over that.
Your Realtor will be able to provide more details.
As a side note, I really wouldn't recommend that a first-time homebuyer pursue short sales. They can take a long time and involve a lot of anxiety, difficulty, and pain. It's important you understand the downsides before proceeding. Again, though, discuss it with your Realtor.
Hope that helps.