Thank you for the post.
The following knowledge base article will assist with the conversion process:
Afterwards, I would suggest reading the entire FileMaker Server 13 Getting Started Guide; however chapter six specifically covers “Upgrading or moving an existing installation”:
“Now before doing so how I can set up a test environment without putting license key, as we have only one server license.”
FileMaker Server 10 and FileMaker Server 13 use different license keys and can be installed and running on the same network at the same time while migrating from 10 to 13.
FileMaker (current version) will convert .fp7 files. In most cases without problems (we've had some minor graphics issues with custom buttons).
Just make sure to have a 'good' backup before converting - and test all Your important processes/functions and printing, before going production.
That said, overworking makes sense. i.e. because of the classic theme (helpfull, but the slowest..) and much more. Reading the links provided by TSFalcon are a 'must'
Testing: You can use one of the clients as temporary 'host' and test in a smal group. Testing locally with only one client is also a first step