It's fairly easy to make a backup of the database and lateron import it into the new server. I walk you through:
Go to your hosts cPanel or equivalent and choose phpMyAdmin. It may be that phpMyAdmin is hidden under the mySQL Databases button,
In the left hand column select the database that holds B2evolution,
Hit the 'Export' tab
In the export menu leave all settings as is, but export the database as 'zip' or 'tar.gz'.
That's all down to personal flavour. You can hit the 'Save to Disk' checkbox and it will present a save dialog, or don't check it and it will present a downloadable file. Afterwards hit the OK button.
A point of warning here. If the database is large these operation *may* lead to an error. That is caused by a time out, meaning the operation takes longer than the 30 odd seconds it is allowed to. No worries. If so you must export the database in two or more parts.
On the new server simply reverse the operation:
First you create a new database with the same name. On a shared host it will do the prefix automatically. In my example the prefix is 'hemmi3_' I created a database called 'b2evolution' and finally it is named 'hemmi3_b2evolution'.
After selecting the database in the left hand column, hit the 'Import' tab.
Simply hit the 'Browse...' button to search the file in your filesystem.
The file may be too large for the upload. The Maximal file size is often limited to 2Mb. (50Mb on my server). In that case you can use an external tool to cut the database in pieces or export the database (in the previous step) in smaller bits.