1. Once your new hosting account is ready upload all files and database 
to their server. Make sure all your database connection is working.
 Put your current site in maintenance mode, thus prevent changes made to 
'old' server which will be abandoned soon.
2. Change domain nameserver to point to the new hosting nameserver.
3. Wait for domain propagation to complete. Test out the new hosting & website.
4. Close your old hosting account, and ask for prorated refund. Yes you can do 
this if your contract haven't ends.
5. Happy website transfering..