One method, assuming you are using PHPMyAdmin.
1) On your 'old' server open PHPMyAdmin then select the 'old' database.
2) Go to 'EXPORT'.
3) Click box ' Save as file' and boxes 'Structure', 'add drop table', 'add auto increment value', 'Enclose table and field names with backquotes' , 'data'.
4) Click 'GO' and save file to convenient place then download to your computer (make a note of where it is saved to).
5) On your 'new' server open PHPMyAdmin then create a new database with exaclty the same name as the 'old' database.
6) Then select the new database (ignore any messages telling you that there are no tables).
7) Select SQL then using 'Location of the textfile' locate the file you saved in 4) and upload this file. This should create all the tables and populate them with the original data.
Hope this helps.
There are complications if you cannot create your own databases using PHPMyAdmin, or if your new server auotmatically puts prefixes to table names, but these can be overcome. Try the above first.