Completely backup your Site files (public_html folder) using Any FTP (like CuteFTP, FileZilla etc). Take backup of DataBase using phpMyAdmin or MyBB's ACP tool:
ACP > Tools & Maintenance > Database Backup
You may also use cPanel tools "Backup":
After entering "Backups" click on "Generate New Fresh Backup" button. Your site files as well as Database will be backed up in server. Choose the option to make the backups available for download. Next downloas 2 files : the DB and the site backup.
Now open your new host's cPanel. Go to databases and create a new database with the user permitted to access.
Note down the database name, database host name, database user name and password as you will be requiring this info during MyBB setup.
Now download the MyBB package from the following link:
http://www.mybb.com/download/latest
extract the files and upload to your new host's "public_html" folder using FTP software.
Properly CHMOD the files as directed in guidelines of MyBB. You can CHMOD the files using FTP software also by navigating in the server files and right-click > set permission.
Quote:CHMOD the following files to 666 (ie, make sure that PHP can write to them).
./inc/settings.php
./inc/config.php (you must rename config.default.php to config.php first)
./inc/languages/*your_language*/*all files* (optional)
./inc/languages/*your_language*/admin/*all files* (optional)
CHMOD the following folders to 777 (ie, make sure that PHP can write to and execute from them).
./cache/
./cache/themes/
./uploads/
./uploads/avatars/
./admin/backups/ (optional)
Now type in your website url which you have setup during registration trailing with "/install" to your browser address bar and go.
Example:
http://yourdomain.com/install
The MyBB setup will be launched. Follow the simple steps. Enter the database details you have noted down earlier and proceed to the finish.
After finishing the setup you will have a fresh MyBB with blank database.
Now fire-up phpMyAdmin from cPanel, select your MyBB database from left and "Import" button from above.
After successful update you will need to extract your old site's backup files taken earlier. Now upload the extracted files to the same "public_html" and replace all.
Almost done. Now go to ACP > Configuration > Settings and activate all the plugins required.
Well it was a first cut guideline. If you face any problem feel free to ask further and I'll be happy to guide you.