how can I search for tasks due between two dates?
Unfortunately, that's not possible. You can search for "30 days", for example, or for specific dates such as "12 May, 13 May, 14 May, 15 May", but you can specify a "from to" date range.
We will consider adding such option in the future, though and you're welcome to request it in our Votebox: https://todoist.com/Vote/
If anyone is interested, I added the suggestion here: http://todoist.com/Vote/show/8449/