Read this and this for an overview with relevant links
Read this to try to come up with a call-sign that sticks to the rules (coffee may be required)
Search for the call sign you want. If you find it already taken you can search for variations you might like. When you submit your application you can submit up to twenty-five new call-signs in order of priority for the administrators to use if your first choice is not available.
As of 2017-02-12 the application is free (a fee of zero dollars)
Go back here to walk through the electronic process of applying and wait for your application to start being processed
Check here to see how the application is proceeding
Check here for call-signs that are predicted to be approved