Sorry to hear about the problems with your current host, I want to confirm I assume you mean mb not gb for both as 254gb would be absolutely huge...?
It is hard to make a precise recommendation with forums as usage is quite variable, but if you are currently on shared hosting and not having resource usage issues then it would make sense to start there, our Power Plan which you can find here would meet your needs:
http://www.nethosted.co.uk/uk-web-hosting.php and costs £9.95+VAT on a monthly basis, getting cheaper if a longer term is committed to. We do offer a 14 day money back guarantee on our web hosting plans so we're happy for you to try it out and if for any reason you're unhappy we'll refund you.
If you did require an upgrade in future then that would be straightforward enough too!
To move the site over you would need a backup of files and databases, it looks as though your current host use Plesk so it would be a manual process to move over to us using cPanel, but we'll be around to assist you with migration should you need it.
If you have any other questions just let us know.
Regards,
Darryl