You can put two sed expressions together using two different -e options:
sed -r -e 's/""+/"/g' -e 's/^"//' file
Regarding the + sign, in a regular expression it means 1 or more occurrences of the previous pattern, whereas the * sign means zero
or more occurrences. In the previous example the regular expression:
ensures it matches only two or more double quotes (the first plus the second one repeated 1 or more times). In other words it leaves single double quotes
untouched (even if - in this case - changing them with another single double quote
doesn't make any difference).
A good tutorial on sed with a lot of regular expressions examples is http://www.grymoire.com/Unix/Sed.html