That's right. "Yes" to the first part of your question ("would i be able to save (burn) the info to a disk"). Use the mysqldump command to send the contents of one or more databases to a text file. The file will contain the commands necessary to recreate the database. Use "mysqldump --help" for the options.
For the second part ("and then use that to perform the DB transfer to the new server"), the answer is also "Yes". Use "mysql < dumpfile.txt" using the filename that you created on your own server.
Again, "mysqldump --help" is your friend. The exact syntax for importing depends on how you dumped it to begin with. For example, you can include the commands to recreate the database, or you can just dump the statements to create and add records to tables assuming that the db already exists.
If I was doing it, and I wanted to transfer a database to a server which doesn't already have it, I'd dump it with:
mysqldump --databases <databasename> > dumpfile.txt
and import it on the new server with:
mysql < dumpfile.txt
Adding the username and password as required.