OK here are the steps you take to back up your database from the cpanel.
It's really very easy!
Log into your cpanel and DO NOT go to the php myadmin area. Hopefully your home page looks like this in step 1.
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Now after you click the backup link in the above pic, it will take you here and you click the link that says your db's name where I have this arrow pointing. In this case this person has a wowbb software (sorry mybb, I tried to get her to convert!)
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Once you click the link to your db, then you will see the pop up to save or open. You want to SAVE this to your pc and THIS file is what you are going to RESTORE on your new server.
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Here is a video tutorial on some things on how to restore a db on your server. It's from my hosting site's tutorial section but I think if your mysql admin area is the same it will do just like I said to do.
Just scroll down midway to the page and look for restore your database link. There is also a link to back up your db in there if your cpanel does not have the stuff I just showed you.
LINK HERE
You need to make sure your nameservers are changed to the new host once you are all done and you are sure it's working.
Now, remember, when you RESTORE according to the instructions in the video, you will see it takes a long time to propagate the tables. If this finishes and you get a red box with an error and a link saying something to the effect it's too big and to upload bigger db's than the server allows to click it.... don't bother! It all turns to gibberish and only a super egghead can understand that!
At that point you need to upload that file you just took off the old server to your offline directory and call your host and ask them to restore your database for you.
You will then need to MAKE SURE it all works and then get your nameservers changed over to the new host as per their directions to you when you purchased hosting. You should do this as soon as possible so you don't lose any posts or new members.
If you know it's going to take time to do this, be sure to back up from the old host DAILY and OFTEN because the second they make the nameserver change, you cannot get access to that old cpanel again or the old databases.
There is a link in there where you can make a full backup of everything on your server but you probably don't need everything that's in there.
Did you move all your files over? You need to move the files over too.
You can make a full install of mybb if you want and then when you restore that db, it will overwrite everything in there and you should be good to go.