It will take the first pair.
How it works:
We're using the MID function, so we need to define what letter to start at, and how many letters we want.
To find the starting letter, we use the FIND function to find the first " mark. Then we add 1, since we don't want to include the quotation mark.
Next, we have to find the second quotation mark. We use a FIND function again, but we use the optional 3rd argument to define a starting number to look at (since we don't want to find the 1st quotation mark again, we use that number +1). This tells us the overall position of 2nd quotation mark.
However, as the MID function wants length of characters to return, we need to subtract the first number we found to get the correct length. Similar to before, the -1 at end is to exclude the 2nd quotation mark.
If there are multiple instances, we could do some tricks using the SUBSTITUTE function. For instance:
=TRIM(MID(SUBSTITUTE(SUBSTITUTE(A3,"""",REPT(" ",999),2),"""",REPT(" ",999),1),500,1500))
Might work. This formula replaces the 1st and 2nd instance of a quotation mark with a big space. The MID function just takes a nice chunk out of the middle, and trims the extra spaces. If you know the total count of quotation marks (see Hui's recent blog post) you can manipulate this to choose which quotation marks to replace.