In Textmate I can "Select Enclosing Brackets" with
How can I set up a similar keyboard shortcut to select everything between the single / double quotes which encompass the cursor?
In TextMate 2 you can use Ctrl+Option+B (Select Current Scope) to do this. Note this will include the quotes in the selection, which is slightly different from what you have pictured, but if what you want is to replace the contents you can just hit delete, one quote and then type.
Once you have the selection you can also use Ctrl-Shift-' to toggle the types of quotes surrounding the string, very handy.
I'm not sure if these are present in TextMate 1 as well.