If you must upgrade instead of creating a new 3.2 site then:
First upgrade to Moodle 2.0 - put the 2.0 compatible versions of your plugins in the Moodle 2.0 code folder before you do the upgrade. Moodle 1.9 to 2.0 was a BIG upgrade.
If you have a lot of quiz attempts then you might have to upgrade the 2.0 site to 2.1. Moodle 2.1 introduced the new question engine which requires a big conversion of your existing quiz attempts. Otherwise, if you don't have lots of quiz attempts, try upgrade the 2.0 site directly to 2.2. Moodle 2.2 is required before you can go to any of the newer versions, so first worry about getting to 2.2.
When that's done and stable you can tackle the jump to 2.7 (which is required before you can upgrade to 3.2).
You have a BIG upgrade path, so you should do it in phases. Run each of the intermediate versions for a few days/weeks before you upgrade to the next version. That way you can fix any issues that you find after each upgrade.
If you have any plugins in 1.9 you'll need to make sure you can find versions compatible with the newer versions of Moodle.
And you will need a new Moodle theme. Your 1.9 theme will not work in the newer versions of Moodle.