Use phpmyadmin to re-upload your MySQL file and just download your files via FTP and just upload them to your new host.
if you're using cpanel: 1. do a full site back up 2. get that backup to your local workstation/laptop 3. contact your new host and get them to restore the backup as simple as that!
Careful with PHPMyAdmin, the newer versions may be ok, I've had issues with export/import in the past...
It depends on What platform the web site is on. And What Server administration software is being used, With cpanel we can create a full backup, if the new host is using same cpanel based servers they can do cpanel - cpanel transfers. you can request them to do this. If you have ssh access you can login and download files. or transfer them to your site directly. Same applies with db's. I have a very good experience in this field, I have transfered huge and big website's some times manually downloading all files and then re uploading. If you need my services you can contact me, if the site less then 100mb I will do this for free(as I always give away freebies to DP members). Regards.
better to do it ourself as when we hire someone, that someone may steal the codes.. sorry to sound a little harsh but it happens
Professionals are more worried about their customer's and their reputation rather then stealing others code.
Well, really appreciate the help here, I am going to change host as you see, my website is getting a lot of downtime. Gonna do the operation myself, hope it goes well. Still waiting for the present host to go up again so that I can get the latest backup files.
That really sucks, to get a back up you have to wait for your host to get their servers back online, is certainly not acceptable.