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Having done this just recently (in fact this is a first post on a new host) here is step-by-step tutorial on how to do that with zero downtime.

1. Backup the database from the old site and import it to the new site. Pay attention to database names.
2. Change your wp-config.php in the old site to point to the database on the new site. Your old site will now run using database based on the new site and once you change the nameservers new site will have the up to date database. (You may want to check with your new hosting if they allow outside database access first. If not, you can skip this step and transfer the database at the end, when the nameserver change propagated)
3. Backup your entire WordPress installation from the old site (this will include all your uploads, themes, plugins)
4. Unpack your WordPress installation to the new site
5. Change wp-config.php on the new site with new database details
6. Change the nameservers to point to the new site
7. That’s all now you only need to wait

Since nameserver propagation can take up to one day, step 2 allows you to have up to date database at the moment it happens. Downside to this is slower site response in the meantime because site still runs from the old server and the database is on the new. But if you do this over the night or periods of low traffic it won’t be that noticeable and you will not lose any comments or other activity on your site.