As a general statement, I would do this:
At the old host:
(1) back up everything on your old server --- both the WordPress files *and* the database. If you don't back up both, then you don't have the whole blog.
Then, at your new host:
(2) install WordPress 1.5. Why? Because your database is in WP 1.5 format, and (I hope I understand this correctly) WordPress 2 made changes to the database.
(3) import your database file into the database at the new host*. This gives you all your posts and settings. If you need help with that, there are plenty of posts about it here in the forums.
(4) If you wish, you can then upgrade to WordPress 2, following instructions you'd find here, in the codex, and at podz' tamba2 site. But, before you do that, you back up everything again. Too many backups is far safer than not enough.
One note: I've just upgraded three blogs from 1.2.2 directly to 2.0.5 (without upgrading to the intermediate 1.5 and 2.0 versions) and it works perfectly. Heck, even the 2.0.5 upgrade.php says, "This file upgrades you from any previous version of WordPress to the latest."
* One of the problems with talking about this stuff is that, so often, things are called the same thing. For instance, mySQL is the database program that runs on the server, and is called "the database". But so are the WordPress files that are in the mySQL database. And so is the backup file that you can make from the files in the database. Enjoy; that's just the way it is! -grin-