I want to change the order of the profile elements. How can I achieve this?

Changing the order of profile elements is not part of the current User Profile plugin functionality. A hard way to rearrange the profile elements can be to recreate them in the order you want. If you do this you loose all the data that users already have inserted to the fields.

This requirement is part of our TODO list. There is an issue regarding this question so you are able to vote on it to increase it's importance.