I used iWeb for years to publish my websites, until it became clear that Apple wasn't going to continue to support iWeb. I've moved most of my sites to Sandvox, but still have a couple to move as I'm writing this. Whether your existing sites are built on iWeb or another application, the process is quite similar.
Publish iWeb Version to New Host
Go to Web Publishing for Sandvox & A2 Hosting and do the Domain Registration, Site Hosting, Setup Host, and Publish steps from within iWeb, except the Setup Host step is a little different, as follow:
Publish to: FTP Server
Site Name: yoursitename (I typically used a short abbreviation here, as a long site name just gunks up the URL. Sandvox doesn't use a site name between the domain name and the rest of the URL.)
Server address: ftp.yournewdomain.com
Username: yourfirstdomainusername (your username that A2 assigned to you via email just after you first opened your account)
This gets your existing iWeb site up. Because of the Site Name iWeb inserts (see above) your old site can exist side-by-side on your A2 domain because the URLs of every page on the old site are guaranteed to be different from the URLs on the new site.
Build Sandvox Site
You can take your time building your new Sandvox site because your old site is still up and running. Once you've built the basics, go ahead and publish it using all of the steps in Web Publishing for Sandvox & A2 Hosting. Unless you build links from the old site to the new site, nobody following links to the old site will know you're building a new one because there won't be any links from the old site to the new (unless you build some into the old site, of course). The only way they would get there is by keying in http://www.yournewdomain.com. Here's a primer on the similarities and differences between iWeb and Sandvox if iWeb is where you're coming from. It sure to make the translation simpler.
Once you're ready to launch the new site, build redirects from each page in the old site to its corresponding page in the new site. If someone follows a link to an old page, it'll automatically send them to the new page instead. Here's more on redirects.
Delete Old Site
…or maybe not. Redirects aren't supposed to need for the old page to still be in place on the internet in order to work properly. So you could delete your old site if you need to save disk space. My A2 account, however, has unlimited disk space, so I haven't bothered to take the old sites down yet.