For the most part moving your site from one host to another is a pretty simple process. You download the site at your current host (or use a local copy you have if it is up to date) and you upload it at the new location. Easy, right? What could go wrong? You'd be surprised as to how many people ask me to help move their websites because they are scared to do it themselves. As it turns out a lot can go wrong, but most of it is pretty easy to fix, it's just a matter of knowing where to look.
One thing that sometimes goes wrong is that somewhere in the transfer process your files permissions get altered. So, for example, if your script needs to be able to write to a file or a directory, but its permissions are changed to not be writable, then of course things aren't going to work and you'll be swimming in a sea of errors. If you get an error about things not being writable, or any error while using a program that writes files to your server (such as letting the user upload a picture) or data to a file, then double check that all of your permissions are set correctly.
Another common mistake is not updating your config files to correctly point to your database.... or worse, forgetting to move your database to your new host. Make sure you have successfully cloned your database on your new server, and that all of you config files point to it. Sometimes localhost will work, and other times it won't, so be aware that you may have to change the data in those files. If you get an error about not being able to connect to your database, then this is most likely the culprit.
Another weird problem I've seen is that some FTP programs like to hide files that start with a period, such as .htaccess. Make sure that your program allows you to see these files so they get transferred along with the rest of your site. Sometimes your PHP program will modify your .htaccess when you install it. If you don't transfer this file along with the rest of your website then things will likely break.
Finally, if your website is not just moving to a different host but also a different domain name you may have to do some modifying. Although a lot of files are relational links, sometimes hard links will be thrown in and may cause you to get dead photos or broken links that go to your old website. Sometimes it's just a matter of changing your URL in a config area, and other times you have to manually change every place you've put in a hard link. Just be aware that you should change them or your page will be less user friendly.
Usually transitioning from one host to another is a very smooth process. If something does go wrong, look for the warning flags in this article so you have a better idea of where to start to fix the problem. Figuring out what is wrong usually takes longer than the actual fix. Best of luck in moving your PHP website to it's new home!