I want to do some practice with deploying changesets. I have sighed up for 30 days trial of Professional edition here http://www.salesforce.com/ap/crm/editions-pricing.jsp I created 1 developer sandbox, but neither in production or sandbox I cannot create apex class. On my profile or permission set there is no permission 'Author Apex' at all. In developer console 'Create new -> Apex class and trigger' buttons are disabled. Is there any way to develop apex for free somewhere else than developer edition?
Professional Edition does not include Apex. You can get a free Developer Org (or more than one) at the Salesforce Developer Site where you have full access to Apex, VisualForce and most other features like Workflow, Process Builder and case management.
Alternatively sign up for Enterprise or Performance edition on a 30 day trial as that will include Apex and Visualforce.
To deploy Apex code into Professional Edition you would need to use packages or AppExchange Apps rather than create in a Developer Org and deploy.
Deploying Apex via Change Sets is not intrinsically different than deploying other components apart from the need to have Test Classes with Code Coverage above 75%