I'll start with ....... be sure to backup your database before you do this, You can’t say I didn’t tell you.
After you have all your settings correct, avatars, attachments, theme, directories etc etc .. updated. You can check with the repair settings tool as well. Some things you can not search and replace as they are stored in serialized arrays, and if you change them they will break. So thats why you need to get the site running proper first.
Run the empty unimportant logs function from maintenance.
Now updating .. this is not necessarily how I would do things, but it should fix most of the headaches with the change over. Assuming you have phpmyadmin or adminer then open the db and run each of the following (obviously use your site name and assuming your tables are prefixed with elkarte). Not tested of course but should work.
UPDATE [icode]elkarte_messages[/icode] SET [icode]body[/icode] = replace(body, "http://www.yoursite.com", "https://www.yoursite.com")
UPDATE [icode]elkarte_personal_messages[/icode] SET [icode]body[/icode] = replace(body, "http://www.yoursite.com", "https://www.yoursite.com")
UPDATE [icode]elkarte_user_drafts[/icode] SET [icode]body[/icode] = replace(body, "http://www.yoursite.com", "https://www.yoursite.com")
UPDATE [icode]elkarte_members[/icode] SET [icode]website_url[/icode] = replace(website_url, "http://www.yoursite.com", "https://www.yoursite.com")
UPDATE [icode]elkarte_members[/icode] SET [icode]signature[/icode] = replace(signature, "http://www.yoursite.com", "https://www.yoursite.com")
There maybe some other areas to target, but that should be the bulk of the trouble. I'm not sure what to do about the sessions table, I'd probably just empty it so things start off fresh.