First, access your old Hosting/cPanel and navigate to the folder of your hosted domain (forum). Select all the files inside the public_html folder, zip them, and download the zip file locally. Then, upload this zip file to your new hosting. Next, go back to the old cPanel, navigate to your Databases, and use phpMyAdmin to access your forum's database. Export the entire database as an SQL file. On the new hosting, create a new database, access it through phpMyAdmin, and upload the SQL file you previously downloaded. If the domain remains the same, everything should work seamlessly.
P.S
Don't forget to update the config file of your forum with the new database credentials.