If you're running the latest WordPress, try this:
1. Install a blank WordPress on your new host.
2. On the old site, go to Manage->Export and export your blog contents.
3. On the new site, go to Manage->Import->WordPress and import your blog contents from the file you got from step 2.
4. Copy over your themes and plugins and any other customizations you have from the old host and put them on the new host.
This is the easy way, but it only keeps the posts and comments and such. Any configuration you have is lost. This is not always a bad thing, it cleans out the database in the process. Using a dump of the database via phpMyAdmin keeps everything, including any crud lying around in there that you may not want.
You didn't say what part of the codex you read, so I'll point out the right part, just in case:
http://codex.wordpress.org/Moving_WordPress#Moving_WordPress_to_a_New_Server