Unless you have server admin/root access at your current host you will have to move them manually. If your current H-Sphere host has HSphere Stats available to you (most do), you can do a search using HSphere Stats and find all accounts complete with databases, email accounts, domains etcetera including usernames/passwords. If you then open your account at the new host and recreate your plans and accounts, emails, and databases you will be ready to move the content.
You can use the "archive" function in File Manager to create a gzipped tar file of domains/folders (leaving them in the account) on each old account by clicking the "folder" icon to the left of each domain/folder. You can then go to your new host->new account end-user account-> File manager and use WGET to pull these gzipped tar files into the account and then uncompress by selecting the file. This is very quick.
Make dumps using phpMyAdmin on the old accounts of all MySQL databases and then import via phpMyAdmin on the new accounts.
I'm looking for a new H-sphere reseller provider - they should offer:
Linux accounts with php4 and php5
stable - good uptime
Though I do not have any personal experience, I do think real-ware.com provides that. You might want to check with them regarding this. Couple of friends of mine are their resellers and they have only good things to say.
I do not see anywhere on their site that suggests they use H-Sphere and/or have multi-platform hosting or reseller hosting for that matter. Actually they offer very little info as far as hosting and seem to be a "partnering" solution?
Eirca has a migration service that they offer for H-Sphere but I recently tried to migrate to their service and the support for this process was a complete failure. Don't chose Eirca based on their site offering this service.