Changing web hosts shouldn't affect your SEO if done properly. Assuming all you are doing is moving the site to a new web host (i.e. not changing domain name or urls):
- Get your new hosting
- Upload the website to your new hosting account
- Get the DNS (domain name server) info from the new hosting account
- Go to your domain registration account
- Enter the new DNS info
- Stock answer is 24-48 hours for all domain name servers to "catch up"
- Once you have confirmed everything is ok at the new location (make a little change to a page or put a comment in the coding to make sure you are seeing the new location) cancel the old hosting
Using this method the move will appear like nothing has changed.
Also, if there is a problem you just change the DNS info back to the old location till you figure out what went wrong.