Web hosting doesn't last forever. Maybe you have a host you've been with for years and years but it's just time to move because you've outgrown them and the initial plans you had for your business when you started with them. Hosts that you've never had a problem with before can sometimes run into problems that are beyond their control but cause their customer service to slip and not be as good as it was before. No matter your reason for leaving your current host, you should consider these things to do as they could easily be overlooked.

Backups

You can never have too many backups. First thing you do before anything else is to create a full backup of your website. This can be easily done using your web host's control panel.

Make sure your email accounts are created over at the new host before you change the name servers. Don't forget email forwards. In the busyness of moving files and getting the website working, things like email accounts and forwarding tend to be forgotten especially if they are accounts you don't use often.

Service Overlap

Don't wait till the last minute to move. Make sure there is some overlap in service between the old and new hosting account, preferably a week or more. Sometimes an ISP hasn't updated or refreshed yet and will continue sending traffic and email to your old site. You want to be able to log in to the old host to check for those straggler emails.

Databases

This is rather unlikely especially when your website is powered by a database. The more likely scenario would be you didn't upgrade the script to reflect the changes. It isn't difficult to move databases but if it's not something that you are comfortable with then get someone else to do it for you.

Page Redirects

Make sure your htaccess and cPanel redirects are re-setup properly. Often, people overlook the htaccess altogether because it may have been hidden from them or the new web host already has an htaccess file created. But just by having an htaccess file doesn't mean your redirects will work automatically because chances are, the one at the new web host is just an empty file. If you have any 301 redirects, you need to open up the htaccess file from your old web host, copy and paste it to the new web host.

Different web hosts have different server setups. If you're moving a cPanel web host to another cPanel web host, this isn't usually an issue. But if you move from one control panel to another then you won't have as smooth a transition and it will be more work. If you're running scripts, check to make sure your scripts are referencing the correct paths. If you're changing the domain of your website as well, check your images and links too.

Files Outside Your Public_html Folder

These files are not public and so they are easy to forget. You may have stored some backups, drafts or anything else you wanted to keep for later in here so be sure to check it and not leave it there.

Seems obvious and easy to spot but not all scripts are in the forefront and forms are one of those. Maybe you haven't installed the form script on the new server or they have a different form mailer. Check that all your forms are working especially the contact forms.

The bigger your site is the better chance there is something that you'll miss. It might be a good idea to take a screenshot of your control panel file manager just to be sure you don't miss anything. Take inventory of the scripts on your site. It's also a good rule to keep this inventory up to date so that as you build your website you have this available next time you do it. All the best on your move.