There are several options for doing what you want to do. Using
mysqldump is certainly one option, but I'm not sure if you need 'root'
access to the server in order to install this tool. Certainly you'll
need shell access, and sometimes with shared hosting enviroments this
isn't provided. The way I've done it in the past is using the PHP and
MySQL management tool, phpmyadmin.
First download the latest phpmyadmin 2.5.3 from http://phpmyadmin.net.
Then unzip/untar the files into a directory called phpmyadmin on the
server containing the database info you want to transfer. Edit the
configuration file provided with phpmyadmin with your username and
password, and you should be good to go.
Next, navigate to the phpmyadmin directory with a web browser, and
click on the name of the database you want to transfer (on the left).
Now click on the tab at the top called 'export', make sure the data
button is checked as well as the structure button. Now click 'go'.
What you're provided with is all the data in your database in the
window, which you can copy and paste into the new server. Phpmyadmin
also gives you the option to save this information as a file, which
might be useful for bigger databases.
To import this information to the new server, follow the instructions
above for installing phpmyadmin and click on the 'SQL' at the top of
the phpmyadmin screen. Paste your database data into the big box, or
locate the file holding the data and upload it.
As a word of caution, you'll want to password protect this phpymadmin
directory, or delete it after you are done with these tasks so as to
make it inaccessible to anyone but yourself.
If you need any clarification, please don't hesistate to ask for a