As far as I know, you won't be able to re-direct people to your new URL, unless you keep the old host and re-direct to your new URL. You might want to keep your old host for a month or two for this purpose.
Buy a domain name. If you run into problems with your new host, you can move the domain to a new host. That way you won't be in the same fix that you find yourself in now.
Re-submit your site to the search engines, after you move to a new host. More and more search engines will not index a site on a free server. A lot of them are purging their indexes of free hosted sites anyway.
I've heard good things about www.5dollarhosting.com. I don't have any experience with them. Buyer beware.
Managing your site is usually easily done with FTP software on a pay host. FTP software can be found for free on the web.
Good Luck.
Dan