Hi PlateSpin,
It's not too difficult to move and shouldn't involve much in the way of downtime, essentially you need to:
1. Ensure you have the ability to manage your domain name. You need to be able to login to a domain management area somewhere (normally with the company you registered your domain name with) - once logged in you should have "Change IPS Tag" and "Change Nameservers" options.
2. Login to your webspace and make a complete backup of all the files to your local PC.
3. Find a new provider and set up a hosting account - ISPReview run a sister site called over at
http://www.host-review.co.uk with hosting company reviews.
4. Your new hosting provider will provide you with their "Nameserver" addresses, this is where you need to login to your domain management tool and modify the Nameserver addresses accordingly.
5. Login to your new webspace and upload all of the files from the earlier backup.
6. Login to your new hosting control panel and create the required email addresses/aliases.
7. Sit back and wait! Nameserver changes can take up to 24 hours to propagate, in the run-up to this change Pipex will still be hosting your website and emails, once the change occurs to the Nameservers the new company will automatically take over the running of the website and emails.
I mentioned the "IPS Tag" earlier, changing this Tag changes essentially changes the company who look after the billing and overall control of your domain name. If the domain was registered through Pipex, the IPS Tag will be pointing to them. Simply changing the Nameservers still gives Pipex billing/renewal control over the domain (even though they will not be hosting your website and emails)
If you do not wish this to be the case, you should be able to transfer your domain to the new hosting company. Simply ask the new company what their IP Tag is, change it in your domain control panel and after a few hours the new hosting company will take overall control and be your new point of contact when it comes to billing/renewing.
I hope the above makes sense - let me know if you need further clarification. Also, the above isn't an exhaustive list but should give you a basic idea of the processes required and the steps to take.