Both are MySQL and you have FTP and telnet.
That is good.
Read the MySQL manual about the 'mysqldump' program.
this utility comes with your mysql installation and lets you dump the database
structure and data into a textfile.
This textfile will contain the SQL queries that create the tables in your database and insert all the data into them.
Run something like this on your 'source' server:
mysqldump --add-drop-table database_name > database.dump
(you may need to enter username and password for security, check the manual)
Then upload the 'database.dump' file to your new server, and run a command like this:
mysql database_name < database.dump
again you may need to enter username and password for security.
Check the MySQL manual for more info