I am moving my site to a new server and upgrading to 1.3.8a. From what I think is 1.3.6. My intention is to do this:

backup all site files and database
do a fresh install of 1.3.8a on the new server
install latest versions of modules (image manager, etc.)
test new site functioning
restore old database to new server
make appropriate changes to site (css, templates)

Is this the way to go? Can someone point me to some detailed description/s of exporting the existing database in the form required, and restoring it to the new server. I think I have it, but I really want to be sure.