When you change web hosts you'll likely have a little down time for at least some of your website visitors.
That is because it takes some time for your domain to propagate to the new dns servers around the world.
If you have a database based website then close the website temporarily down to new replies if it's a forum.
Then while your site is down for the move, start changing your dns settings for your domain name to your new web hosts dns server names.
This gives your domain time to propagate to the new server while you're uploading files and databases to the new web server.
This will decrease the amount of downtime your website sees. But you will still get a little downtime when moving your website to a new web host.